The facility with delay penalty function of large value may reduce the number of waiting passengers there, and the place with delay penalty function of low value can reduce passenger antipathy to waiting. In other words, the value of functions can decide the place where the operators prefer passengers to wait.
Therefore, such a flow control strategy is a nonequilibrium solution. In other words, the flow control rate among successive stations or successive flow control periods can be unbalanced.
As a result, the linear delay penalty functions are proposed and they will be discussed in Model II. Model II: equilibrium flow control model: in Model II, the delay penalty functions are set as two linear functions, and , respectively, in which and are parameters. Compared to constant delay penalty, the linear delay penalty with quadratic objective function in Model II can lead to an equilibrium solution for both time and space; the proof is shown in Appendix A.
Model I and Model II are both integer nonlinear programming models. They both have the same variables and constraints. The complexity of the models are determined by three preset parameters , , and on the considered line corridor, which are the number of stations, the number of trains, and the number of flow control periods, respectively. Table 3 lists the maximum number of variables and constraints.
To illustrate the problem, an example with 10 stations and 10 trains in 10 periods is provided. There are integer decision variables, 1, intermediate variables, and 1, constraints at most. With the prespecified parameters increasing, the number of variables and constraints will show a geometric increase. Furthermore, because the decision variable refers to the number of controlled passengers with high precision, the high-precision result is difficult to realize in reality.
A near-optimal solution is also acceptable as long as the solution is close to the global optimal solution and achieves satisfactory. It is also important to develop an easy methodology which can provide a good solution within acceptable computation time. Therefore, we develop a heuristic algorithm, namely, ISA algorithm, to obtain a feasible solution for the passenger flow control problem.
To analyse the advantages and disadvantages of optimized flow control strategy, expect for the total passenger delay including passenger left-behind delay and flow control delay , more performance indicators should be considered.
Equilibrium is a flow control characteristic that is difficult to describe quantitatively. To discuss the temporal and spatial traits of passenger flow control strategy, spatial equilibrium indicator and temporal equilibrium indicator are proposed, which reflect the equilibrium of strategy.
Spatial equilibrium indicator is used to describe the deviation of passengers who enter different stations in the same period. The construction of the performance indicators involves the standard deviation in statistics.
The average controlled passengers at different stations in the same period are calculated by equation The spatial equilibrium indicator in each period is shown in equation Then, the spatial equilibrium indicator is derived as shown in equation 27 :. Temporal equilibrium indicator implies the fluctuation of passengers who enter the same station in different periods.
The variation of controlled passengers can be shown as the gap between the average number of controlled passengers in former and latter periods and the number of controlled passengers in present periods equation Then, the temporal equilibrium indicator is derived as shown in equation 29 :.
The two indicators are smaller, and the flow control strategy has more equilibrium, which means the deviation of controlled passengers is smaller at different stations and the fluctuation of that is smoother in different periods.
In the proposed model, the decision variables are the number of controlled passengers on each station at each period, which is a series of integers with a finite range the lower bound is 0 and the upper bound is the number of total arrival passengers , and the feasible solutions may be close and even similar.
An algorithm is needed to randomly search the feasible solutions in a local area and also avoid falling into the local optimal solution.
Therefore, simulated annealing SA algorithm is suitable, which is a random local search method with different search strategies, acceptance, and stop conditions, which is able to avoid falling into poor quality local optimal solution by accept nonimproved solution with certain probabilities.
To guarantee the convergence of the algorithm, the accept probability is related to the temperature, with temperature cooling down and the acceptable probability goes to 0 [ 26 ]. SA originated from statistical mechanics, which is a solution method that is based on the physical process of simulated annealing.
It was firstly proposed to use in combinatorial optimization problem by Kirkpatrick et al. The method has strong applicability and can obtain solutions arbitrarily close to the optimum [ 28 ]. In flow control problem, the decision variable is the number of controlled passengers and the intermediate variables are also the number of passengers at each area, so the optimal result is difficult to find in large passenger flow. A number of passengers at some place have relationships with each other, a decrease in one place will lead to an increase in the other, so the range of search in each iteration is difficult to decide.
For example, when the waiting passengers at the entrance entered station, the number of waiting passengers decreased and the number of entered passengers increased, and the amount of change is the same. To adapt to the flow control problem, an improved simulated annealing ISA algorithm with the random disturbance operator is constructed.
After the algorithm introduction, the parameters will be explained in Table 4. The pseudocode of the ISA algorithm is shown in Algorithm 1. The passenger demand, basic operation constraints, and basic algorithm parameters are set as the input datasets.
Firstly, the algorithm will be initialized in Step 1, a solution without any flow control strategy will be given as the initial solution, and the temperature is set as the start temperature. Secondly, the current solution will be applied a random disturbance, and a new solution is produced in Step 2.
Thirdly, the new value of objective functions will be calculated in Step 3 by equations 7 — Then, in Step 4, the new solution will be accepted as the best solution if it is better than the current one, or the random bad solution accepted condition is existence. The temperature will be cooled down with the cooling rate, if the temperature is higher than the end temperature, and if not, the algorithm will be stopped.
Finally, the best flow control strategy and the best objective value will be put out. Specifically, there are four steps. Step 1: initialization, Step 2: random disturbance, Step 3: result calculation, and Step 4: result acceptance in the ISA algorithm.
Initialization includes algorithm initialization and model initialization. The algorithm initialization sets the start temperature , cooling rate , and end temperature. The model initialization sets the passenger arrival rate , the proportion of passengers at each station , initial flow control strategy , and initial result.
Specifically, the initial flow control strategy does not control any passengers; in other words, is equal to 0. Roulette wheel selection is used to produce a random disturbance operator.
The random disturbance operator changes the strategy by , and then a new passenger flow control strategy can be obtained. Specifically, it will change the number of controlled passengers at each station in each period. To increase randomness, a random parameter is set to note a random number from 0 to 1. To control the disturbance in a reasonable range, the order of magnitude of the number of waiting passengers at the entrance will be calculated at first equation Equation 30 calculates the maximum order of magnitude of waiting passengers at the entrance.
Then, the disturbance is calculated by the equation Finally, combine the current strategy with the random disturbance to get a new passenger flow control strategy. Specifically, the step length of search is decided by the maximum order of magnitude and the precision of disturbance , which can be adjusted by the number of waiting passengers at the entrance and improve the efficiency of the algorithm.
For example, the disturbance precision is 0. If the maximum number of waiting passenger is at period , the magnitude is and the random disturbance operator is Based on the constraints 7 — 20 , the intermediate variables can be calculated, if a passenger flow control strategy is given.
The new passenger flow control strategy that includes the controlled passengers at each station in each period is obtained by step 2 random disturbance , so the intermediate variables in new solution can be calculated and the objective value in new solution can be calculated.
To avoid falling into a local optimal solution in the computation process, the nonimproved solution with nonimproved objective value will be accepted with a certain probability. The nonimproved solution will be accepted if the random nonimproved solution accepted condition exists.
At first, we constructed a small-scale numerical experiment to validate the effectiveness of the proposed single-line synchronous flow control model. In the numerical experiment, a line within one direction includes 5 stations and 4 sections. The study time horizon is 65 minutes — There are 5 trains, and all of them stop at each station. As shown in Figure 4 , trains run from station 1 to station 5, and the headway of all the successive trains is 3 minutes.
The first train departs from the first station at , from which the train location parameter can be obtained. There are five flow control periods, and the passenger flow rate is same in one period Assumption 2 ; different colours of double arrows in Figure 4 imply different flow control periods. Specifically, the number of controlled passengers is the same in one period.
The start time of recording arrival time of passengers at different stations is , , , , and , respectively, which is shown with green arrows in Figure 4. The passenger arrival rate on each station at each period is shown in Table 5. The OD probability matrix for passengers the ratio of passengers from a station to another is shown in Table 6. Train capacity is , and the number of passengers on platform at the beginning of the study horizon and the capacity on platform is shown in Table 7.
The comparison of the results of the two methods is shown in Table 8. The ISA algorithm has iterated over times, and the result almost converges when iterating to times as shown in Figure 5. The total delay of optimized result is , which is more than that of optimal result The relative gap is 2. Besides, the computation time of ISA algorithm is But, the range of search in each temperature of ISA algorithm may be small, which may lead to slower convergence and longer computation time.
The detailed passenger flow control strategies of optimized and optimal solutions are compared and shown in Table 9. Only a small part of the result data is different, and the difference is not more than 5. Therefore, the ISA algorithm has high precision. To demonstrate the effectiveness of the model, the optimized result is compared with the original result without flow control.
The comparison of results is shown in Table In the original result, passengers will not be controlled at the entrance, the flow control strategy is not adopted, and the delay of flow control is 0.
Since many passengers were left behind at the platform because of train capacity limitation, it may lead to safety issues. However, in the optimized result, there is no passenger left behind at the platform because of the flow control strategy, though the passengers flow control delay at the entrance still exists. Specifically, compared with the original result, the total delay is reduced by minutes The flow control strategy is nonequilibrium, since the number of controlled passengers in the first period on stations 3 and 4 are too large, while it is small in other periods at other stations.
The strategy fluctuates wildly in terms of both successive periods and successive stations, so it is difficult to implement in practice and may cause continued shut down at some stations. The waiting time at the platform is the same, which means the number of boarding passengers is the same.
In summary, although the model provides a flow control strategy which can reduce the number of left-behind passengers at the platform and also reduce the total delay of passengers, it is nonequilibrium. To analyze the sensitivity of the parameters in Model I, some different values of parameters are tested, in which the value of is 0. The total delay, flow control delay, and left-behind delay with different parameters are shown in Figure 6.
In the test, the left-behind delay is 0 when , in which no passengers are left behind at the station. Besides, the control delay is 0 when , in which no passengers are controlled at the entrance.
When and , the flow control delay and left-behind delay are balanced well. Compared with the total delay of these tests, the total delay is minimized when. Therefore, it is better to control the passengers at the entrance instead of staying at the platform in the case of oversaturation.
In summary, the total delay is minimum when , which are the best parameters relationship in the Model I. To avoid the influence of the parameters, we set and in constant and linear delay penalty functions of Models I and II, respectively. The comparison of results of two models is shown in Table Compared with the result in Model I, the flow control delay is smaller and the left-behind delay is larger in Model II, and the passenger total delay in constant function test is also smaller than that in Model II.
Because more passengers are left behind at the platform instead controlled at the entrance in Model II, the passenger waiting time at the platform in two models is same, which means the number of serviced passengers is same in two strategies.
There are added In summary, the total delay in Model I is smaller than that in Model II, but the flow control delay is reversed, and the strategy from Model II is more equilibrium than that from Model I no matter if it is on temporal or spatial. The details of passenger distribution at stations in two tests are compared, and the number of controlled passengers, the number of waiting passengers at the entrance and the platform, and the number of left-behind passengers are shown from Figures 7 — 10 , respectively.
Compared with the passenger distribution in the solution of Model I, the controlled passengers at the front stations such as stations 1 and 2 are more and that at the behind stations such as stations 3 and 4 are less in the solution of Model II in period 1.
In other words, the passengers at the front stations are controlled, and more passengers at the behind stations can boarding the train, because the train capacity will be reserved for the behind stations. This strategy avoids the situation that some stations are controlling too many passengers while other stations do not take any flow control strategy, which can relieve the high-impact flow control strategy at behind stations.
Compared with the waiting passengers at the entrance and platform in Model I, the number of waiting passengers at the entrance and platform in Model II is similar between different stations. However, some passengers are left behind at the platform in Model II, due to the weight of left-behind delay is not high enough, which may not be the best parameter relationship in Model II. To analyze the weight of linear functions, sensitivity of parameters in Model II will be analyzed in Section 5.
In summary, considering the equilibrium of waiting passengers at each station and the passenger patience while waiting at the entrance, it s observed that the solution in Model II is better. There are different values of parameters in Model II, which will be tested to analyze the sensitivity of the linear delay penalty functions. In the tests, firstly, the value of is set to 0. The total delay, flow control delay, and left-behind delay with different results are shown in Figure The left-behind delay is almost 0 when , in which almost no passengers are left behind at the station.
Besides, the flow control delay is quite small when , in which few passengers are controlled at the entrance. When , the flow control delay and left-behind delay exist. They are balanced while and. So, when passengers are oversaturated in the system, they should better be controlled at the entrance instead of left-behind at the platform in Model II.
In summary, the total delay is minimum when , which are the best parameter relationships in the Model II. Beijing Subway Batong Line is one of the lines that operated by the Beijing Metro Company, which connects the central city and Tongzhou suburban area in the east. The layout of Batong Line is shown in Figure It is in east-west direction and has a total length of It adopts a fixed six-car group with 1, service capacity.
Except for the terminal station Sihui station , the platforms of all stations on the Batong line are side type instead of island type, which is easy to control the inflow in down direction by setting up the fencing facility at the halls. The arrival time of train 1 at each station of Batong Line is shown in Table The train departure headway is 3 minutes; then we can get the train departure time parameters. The capacity of platform at each station is shown in Table 12 , which is the product of the platform active area and the maximum passenger gathering density.
The platform active area is the passenger usable area that is the total platform area minus the occupied area by the infrastructures, and the data are from the on-site survey. Passenger demand is collected by the Automatic Fare Collection AFC system, which includes the swiping data of passengers. However, the number of entering passengers in peak hours may be reduced because the swiping data are under the original follow control strategy that the operator controlled with their managerial experience.
To recover the real passenger arrival rate at the entrance, the swiping data were magnified 1. Because there are few of passengers alight at intermediate station and the train capacity is limited, the passengers are difficult to board and many passengers are left behind on the platform at downstream stations.
To keep the metro system operation safety, the passenger flow control strategy is necessary to apply. The Batong Line starts to operate at , so we can calculate the number of passengers at the platform by the AFC data. We find that no passengers will be left behind at the platform before because of the sufficient train capacity.
Therefore, there are no passengers at the platform at the beginning of study time horizon. The other necessary parameters of train and line are shown in Table To demonstrate the effectiveness of the Model I and Model II and the ISA algorithm, four experiments are designed with different conditions and parameters, which are shown in Table Experiment A is the original solution without any flow control strategy.
The Model II with best parameters relationship is used in Experiment B, from which can get the best equilibrium result. The Model I with best parameters relationship is also used in Experiment C, which can produce the best nonequilibrium result.
The results of four experiments are listed in Table In experiment B, the Model II is used to optimal the passenger flow control strategy, in which the linear delay penalty functions are ,.
The ISA algorithm is used to solve the model, and it takes The flow control strategy at each station in each period is shown in Figure Compared with the solution A, the total delay is decreased to 80, Besides, the total waiting time of passengers is also saved 80, In summary, the proposed equilibrium passenger flow control model can provide an effective flow control strategy, in which the passenger delay and waiting time can be greatly decreased.
To analyze the difference of the Model I and Model II, the two models with same parameters ; are used in the experiments C and D, which get solutions C and D, respectively. Compared with the solution C, the left-behind delay 3, minutes is sacrificed to reduce the flow control delay 79, Compared with solution B, more flow control delay is used and less left-behind delay is used in solution C, and total delay is decreased 54 minutes.
Furthermore, the computation time is least in solution C. Therefore, the ISA algorithm is efficient. The two equilibrium performance indicators of flow control strategies with solutions B, C, and D are shown in Table 18 , which also include the maximum spatial equilibrium indicator in the different periods and the maximum variation of controlled passengers at different stations in different periods.
For , solutions B and D are smaller than solution C, which means the results of Model II are more equilibrium on spatial and the number of controlled passengers is similar in different stations for each period. Considering the maximum , solutions B and D are smaller, which means the strategy of Model II are still equilibrium on spatial even in the worst period.
For , solution B and D is smaller than solution C, which means the results of Model II are more equilibrium on temporal and the fluctuation of passengers in different periods is smooth. Consider the maximum , solutions B and D are smaller, which means the strategy of Model II is still equilibrium on temporal even in the worst situation.
In summary, the strategy of Model II is equilibrium no matter on spatial or temporal. To analyse the equilibrium of strategy on spatial and temporal in detail, the flow control strategies of solutions B, C, and D at each station in each period are shown in Figures 13 a — 13 c. In solution C, in the same period, the number of controlled passengers at successive stations is quite different, in which some stations control over passengers in a period and some stations just control less than passengers.
That makes the passengers at different stations feel unfair. Furthermore, at the same station, the number of controlled passengers at different successive periods fluctuates wildly, which is nonequilibrium, so the flow control strategy of solution C cannot be used in practice. While, in solutions B and D, the problems in solution C can be solved. In the same period, the number of controlled passengers is similar at successive stations expect for the last two transfer stations because the train have enough capacity at last two stations , which can avoid some stations controlling a lot and most of the stations just controlling no more than passengers in a period.
Besides, at the same station, the number of controlled passengers is rising as time goes on, but the change of controlled passengers is relatively smooth, which is an equilibrium result and easy to carry out in reality. Compared with the solution D, the flow control strategy of solution C is more strictly left-behind passengers are less , so the controlled passengers are more in each period at each station.
In summary, the result of constant delay penalty factor is better than linear one, but the flow control strategy is nonequilibrium, which is difficult to be used in reality. The flow control strategy with linear delay penalty function is equilibrium. It not only balances the number of controlled passengers at different stations but also is easy to be used in reality. As shown in Figure 14 , the flow control delay, left-behind delay, and total delay of each station are compared to analysis the delay time on each station of each solution.
Compared with solution A, the total delay and the left-behind delay of station 10 are decreased and the flow control delay of previous stations stations is increased in solutions B, C, and D. Specifically, compared with the result of solution A without any flow control strategy , the total delay on station 10 is greatly alleviated in the solutions B, C, and D with the flow control strategy from the proposed models , which is borne by the previous stations equally.
Furthermore, compared with the result of solution C by the Model I, the total delay is more equally from station 1 to station 10 in solution B by the Model II, in which the station total delay is closer to the average total delay at stations 1, 9, and 10 in solution B. Therefore, the proposed equilibrium flow control model Model II can control the delay of different stations more equal, so that some situations which may lead to extensively long delays for a small number of stations stations 9 and 10 were avoided.
In reality, the Communication University of China station is one of the most stations with left-behind passengers, not only because of the large passenger demand at this station but also because the train loading rate is almost saturated before arriving at this station. Therefore, seven critical indicators will be discussed, which includes four number of passengers indicators the number of controlled passengers at the entrance, the number of left-behind passengers at the platform, and the number of waiting passengers at entrance and platform in each period at this station and three delay indicators the flow control delay at the entrance, the left-behind delay at the platform, and the total delay of passenger travel time in each period at this station.
In Figure 15 a , the number of controlled passengers in solutions C and D fluctuates wildly, which is a nonequilibrium flow control strategy. In solution B, the controlled passengers are rising and then falling down smoothly, which is an equilibrium strategy for real-word. Consider the left-behind passengers at the platform in Figure 15 b , almost no passengers are left behind in solutions B and C, while there are more than passengers are left behind in solution D in study time horizon.
Therefore, the flow control strategies of solutions B and C can effectively avoid the situation that passengers are left behind at the platform. At the entrance Figure 15 c , in solution D, over passengers are waiting at the entrance in period , which may lead to the entrance overcrowding. But, in solutions B and C, the number of waiting passengers at the entrance is from to , which is relatively moderate.
Therefore, the solutions B and C can avoid the crowded queue at the entrance. At the platform Figure 15 d , over passengers are waiting in periods 17 and 24 in solution D and over passengers are waiting in some periods in solution C, which may lead to the platform overcrowding. While, in solution B, no more than passengers are waiting at the platform in each period. Therefore, the solution B can avoid overcrowding at the platform. Then, as shown in Figures 15 e — 15 g , three kinds of delay in different solutions are compared.
As time goes on, the left-behind delay and the total delay of solution A increase rapidly and over 10, minutes in periods 24 and 25, which is far greater than the value in other solutions the delay is not exceeding 2, minutes. Therefore, compared with the solution without any flow control strategy solution A , the solution from the proposed models solutions B, C, and D can reduce the passenger left-behind delay and the total delay greatly at specific stations.
The situation that may lead to extensively long delays for a station can be avoided. Furthermore, compared with the flow control delay or the total delay in solution C, the results in solution B is smaller in periods It means that the flow control strategy from the Model II with the best parameters relationship can reduce the passenger left-behind delay and the total delay at a specific station better than that from the Model I with the best parameters relationship.
In summary, the equilibrium passenger flow control strategy of solution B can not only avoid the overcrowding situation at the entrance and the platform but also be easily implemented in practice.
Passenger flow control is one of the most effective methods to alleviate passenger crowd in station for an oversaturated urban rail transit system, especially when the train frequencies cannot be further increased due to the signal system. In this study, an optimum passenger flow control model was constructed for a line, in which the line is unidirectional or the line is bidirectional, but the inflow with opposite directions can be controlled, respectively.
The objective function was to minimize the total passenger delay, which consists of the delay of passenger flow control at the entrance and the delay of the passenger left-behind at the platform. The academic contribution of the model is that flow control strategies are controlled by delay penalty functions.
By applying different forms of delay penalty functions, i. Based on the characteristics of the model, the ISA algorithm was proposed, which was based on the random disturbance operator and could solve the model with high accuracy and high efficiency.
Compared with the original result without any flow control strategy, the total delay decreased minutes Therefore, the flow control strategy can decrease the total delay. Compared with the nonequilibrium solution by Model I, the equilibrium solution by Model II can balance the number of controlled passengers at successive stations and in successive periods.
Meanwhile, the sensitivities of the parameters in Model I and Model II were analyzed, and the best parameters relationship could be confirmed: the total delay was minimum when in Model I and in Model II. A real-world case from Batong Line railway corridor in Beijing urban rail transit system was used to further compare the strategies of two models and test the applicability of the algorithm in practice.
The best parameter relationship in Model I and Model II and the two models with same parameters were tested and discussed. The result showed that the result in Model I was better than Model II no matter in total delay time or in total waiting time, but the flow control strategy of Model I was nonequilibrium, which is difficult to be used in reality. The flow control strategy of Model II was equilibrium, which can balance well between successive stations and successive periods.
The flow control strategies with different solutions in Communication University of China station were compared. The result shows that the optimum equilibrium flow control model not only avoids the overcrowding situation at the entrance and platform but also balances the number of controlled passengers in successive periods.
This can significantly improve the flow control strategy and facilitate the implementation. It concludes that the model is effective for practical use. In the future, the research should be studied in the following aspects: i the passenger arrival rate and the proportion of passengers from original station to the destination station should be more accurate. Therefore, more types of fairness should be discussed, such as proportion fairness.
If the number of controlled passengers is proportional to the number of arrived passengers, it is another fairness and may consistent with the reality. Before we explain the significance of quadratic objective function in Model II, the difference between linear function and quadratic function in resource allocation problem will be discussed. Firstly, a simple example about resource allocation problem is proposed.
On spatial allocation, the temporal allocation is ignored, which means that the analyzed case is in a period of time. There are stations on the line with corridor transit capacity in one direction. We assume that the number of arrived passenger at station is , which is no less than and their destination is station. To meet the oversaturated demand with limited capacity, the resource should be allocated to each station.
A linear model LM is proposed and shown as follows, which has a linear objective function with two constraints equations A.
And, the best result is obvious, , with the solution that any meet. The allocation result is not decided by the objective function. For example, a solution and another solution are both the best solutions of LM, but the resource allocation are different and nonequilibrium:.
A quadratic model QM is proposed, which has a quadratic objective function with constraints A. The QM is convex, because and. To solve the QM, the constraint A. Then, the constraint A. Setting for each flow gives. Substituting for the to equation A. Journal overview. Special Issues. Received 10 Dec Revised 03 Dec Accepted 06 Feb Published 25 Feb Abstract When passengers are oversaturated in the urban rail transit system and a further increase of train frequency is impossible, passenger flow control strategy is an indispensable approach to avoid congestion and ensure safety.
Introduction 1. Background With the development of urban rail transit, the number of passengers is increasing and the passenger demand has been far exceeded the transportation capacity of the railway system. Figure 1. The second equilibrium condition means that in equilibrium, there is no net external torque to cause rotation about any axis. The first and second equilibrium conditions are stated in a particular reference frame. The first condition involves only forces and is therefore independent of the origin of the reference frame.
However, the second condition involves torque, which is defined as a cross product,. Therefore, torque depends on the location of the axis in the reference frame. However, when rotational and translational equilibrium conditions hold simultaneously in one frame of reference, then they also hold in any other inertial frame of reference, so that the net torque about any axis of rotation is still zero.
The explanation for this is fairly straightforward. From our study of relative motion, we know that in the new frame of reference. In the final step in this chain of reasoning, we used the fact that in equilibrium in the old frame of reference, S , the first term vanishes because of Figure and the second term vanishes because of Figure. Hence, we see that the net torque in any inertial frame of reference. The practical implication of this is that when applying equilibrium conditions for a rigid body, we are free to choose any point as the origin of the reference frame.
Our choice of reference frame is dictated by the physical specifics of the problem we are solving. In one frame of reference, the mathematical form of the equilibrium conditions may be quite complicated, whereas in another frame, the same conditions may have a simpler mathematical form that is easy to solve.
The origin of a selected frame of reference is called the pivot point. In the most general case, equilibrium conditions are expressed by the six scalar equations Figure and Figure. For planar equilibrium problems with rotation about a fixed axis, which we consider in this chapter, we can reduce the number of equations to three. The standard procedure is to adopt a frame of reference where the z -axis is the axis of rotation.
With this choice of axis, the net torque has only a z -component, all forces that have non-zero torques lie in the xy -plane, and therefore contributions to the net torque come from only the x — and y -components of external forces. Thus, for planar problems with the axis of rotation perpendicular to the xy -plane, we have the following three equilibrium conditions for forces and torques:. In Figure , we simplified the notation by dropping the subscript z , but we understand here that the summation is over all contributions along the z -axis, which is the axis of rotation.
In Figure , the z -component of torque. The angle. When using Figure , we often compute the magnitude of torque and assign its sense as either positive. In Figure , net torque is the sum of terms, with each term computed from Figure , and each term must have the correct sense. Similarly, in Figure , we assign the.
The same rule must be consistently followed in Figure , when computing force components along the y -axis. View this demonstration to see two forces act on a rigid square in two dimensions. At all times, the static equilibrium conditions given by Figure through Figure are satisfied. You can vary magnitudes of the forces and their lever arms and observe the effect these changes have on the square.
In many equilibrium situations, one of the forces acting on the body is its weight. In free-body diagrams, the weight vector is attached to the center of gravity of the body. For all practical purposes, the center of gravity is identical to the center of mass, as you learned in Linear Momentum and Collisions on linear momentum and collisions.
Only in situations where a body has a large spatial extension so that the gravitational field is nonuniform throughout its volume, are the center of gravity and the center of mass located at different points.
In these situations, the center of gravity is identical to the center of mass. Therefore, throughout this chapter, we use the center of mass CM as the point where the weight vector is attached. Recall that the CM has a special physical meaning: When an external force is applied to a body at exactly its CM, the body as a whole undergoes translational motion and such a force does not cause rotation.
When the CM is located off the axis of rotation, a net gravitational torque occurs on an object. Gravitational torque is the torque caused by weight. This gravitational torque may rotate the object if there is no support present to balance it. The magnitude of the gravitational torque depends on how far away from the pivot the CM is located.
Passenger cars with a low-lying CM, close to the pavement, are more resistant to tipping over than are trucks. If the center of gravity is within the area of support, the truck returns to its initial position after tipping [see the left panel in b ]. But if the center of gravity lies outside the area of support, the truck turns over [see the right panel in b ].
Both vehicles in b are out of equilibrium. Notice that the car in a is in equilibrium: The low location of its center of gravity makes it hard to tip over. If you tilt a box so that one edge remains in contact with the table beneath it, then one edge of the base of support becomes a pivot.
As long as the center of gravity of the box remains over the base of support, gravitational torque rotates the box back toward its original position of stable equilibrium. When the center of gravity moves outside of the base of support, gravitational torque rotates the box in the opposite direction, and the box rolls over.
View this demonstration to experiment with stable and unstable positions of a box. A passenger car with a 2. Where is the CM of this car located with respect to the rear axle? We do not know the weight w of the car. All we know is that when the car rests on a level surface, 0. Also, the contact points are separated from each other by the distance.
We also know that the car is an example of a rigid body in equilibrium whose entire weight w acts at its CM. The CM is located somewhere between the points where the normal reaction forces act, somewhere at a distance x from the point where. Our task is to find x. Thus, we identify three forces acting on the body the car , and we can draw a free-body diagram for the extended rigid body, as shown in Figure. We are almost ready to write down equilibrium conditions Figure through Figure for the car, but first we must decide on the reference frame.
Suppose we choose the x -axis along the length of the car, the y -axis vertical, and the z -axis perpendicular to this xy -plane. With this choice we only need to write Figure and Figure because all the y -components are identically zero. Now we need to decide on the location of the pivot point. We can choose any point as the location of the axis of rotation z -axis. Suppose we place the axis of rotation at CM, as indicated in the free-body diagram for the car.
At this point, we are ready to write the equilibrium conditions for the car. The first equilibrium condition, Figure , reads. This condition is trivially satisfied because when we substitute the data, Figure becomes. The second equilibrium condition, Figure , reads. When the pivot is located at CM, the gravitational torque is identically zero because the lever arm of the weight with respect to an axis that passes through CM is zero.
The lines of action of both normal reaction forces are perpendicular to their lever arms, so in Figure , we have. With this information, we write the second equilibrium condition as. We can now write the second equilibrium condition, Figure , explicitly in terms of the unknown distance x :. Here the weight w cancels and we can solve the equation for the unknown position x of the CM. The answer is.
Choosing the pivot at the position of the front axle does not change the result. The free-body diagram for this pivot location is presented in Figure. For this choice of pivot point, the second equilibrium condition is. The answer obtained by solving Figure is, again,.
This example shows that when solving static equilibrium problems, we are free to choose the pivot location. For different choices of the pivot point we have different sets of equilibrium conditions to solve. However, all choices lead to the same solution to the problem. Solve Figure by choosing the pivot at the location of the rear axle. Explain which one of the following situations satisfies both equilibrium conditions: a a tennis ball that does not spin as it travels in the air; b a pelican that is gliding in the air at a constant velocity at one altitude; or c a crankshaft in the engine of a parked car.
A special case of static equilibrium occurs when all external forces on an object act at or along the axis of rotation or when the spatial extension of the object can be disregarded. In such a case, the object can be effectively treated like a point mass. In this special case, we need not worry about the second equilibrium condition, Figure , because all torques are identically zero and the first equilibrium condition for forces is the only condition to be satisfied.
You will see a typical equilibrium situation involving only the first equilibrium condition in the next example. View this demonstration to see three weights that are connected by strings over pulleys and tied together in a knot. You can experiment with the weights to see how they affect the equilibrium position of the knot and, at the same time, see the vector-diagram representation of the first equilibrium condition at work. A small pan of mass The maximum tension that the string can support is 2.
Mass is added gradually to the pan until one of the strings snaps. Which string is it? How much mass must be added for this to occur? This mechanical system consisting of strings, masses, and the pan is in static equilibrium.
Specifically, the knot that ties the strings to the pan is in static equilibrium.
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