Method for calculating safety coefficient of spider type overhead working truck based on rigid-flexible coupling model

Abstract

The invention discloses a method for calculating the safety coefficient of a spider type overhead working truck, which belongs to the field of overhead working trucks and comprises the following steps of: firstly, modeling a three-dimensional rigid model; importing the three-dimensional model into ADAMS software, performing Boolean operation merging on the model, and adding constraints and drives; carrying out flexible processing on each section of boom model of the whole vehicle in ANSYS software to obtain an MNF file and outputting the MNF file; re-importing the MNF file into the ADAMS software to replace an original rigid arm support model; carrying out rigid-flexible coupling multi-dynamics model simulation, and obtaining the real-time change condition of the contact counterforce between the supporting leg and the ground by utilizing a post-processing function; wherein the spider type overhead working truck has four overturning lines AB, BC, CD and DA, an overturning moment Mr and a stabilizing moment Ms are calculated by utilizing a formula, and finally, a safety coefficient K is calculated by combining the overturning moment Mr and the stabilizing moment Ms calculated in the step S7; the dynamic safety coefficient of the spider type overhead working truck is obtained through the change of the supporting reaction force of the supporting legs, and the authenticity of stability evaluation can be improved.

Description

Technical Field

[0001] The present invention relates to the technical field of aerial work vehicles, and in particular to a method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model. Background Art

[0002] As a large-scale mechanical equipment, aerial work vehicles have been widely used in many fields such as municipal administration, fire protection, construction, etc. As a manned equipment, aerial work vehicles will seriously threaten the lives of personnel in the event of an accident. Safety issues cannot be underestimated. In the actual production process, there are many cases of accidents caused by stability problems. Therefore, stability is an important indicator for evaluating the safety performance of aerial work vehicles. Since some components of spider-type aerial work vehicles are prone to elastic deformation during operation, it is a rigid-flexible hybrid system. At this stage, the evaluation method of stability still remains at the ratio between the overturning moment and the stabilizing moment calculated by a pure rigid model. This results in overly conservative results, and the entire calculation process is inefficient and costly, and the results deviate significantly from the true value.

[0003] Therefore, based on the multi-rigid body model, the applicant combined the load characteristics of the aerial work vehicle under actual working conditions and established a rigid-flexible coupling dynamic model using ADAMS and ANSYS software. The dynamic stability coefficient was obtained through the real-time changes of the support reaction force of the legs. This coefficient was used as an indicator to evaluate the stability of the equipment to improve the accuracy of the stability evaluation. Summary of the Invention

[0004] In order to make up for the shortcomings of the existing technology, the present invention provides a method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model, which is used to solve the problem of inaccurate stability calculation of spider-type aerial work vehicles in the existing technology.

[0005] To solve the above technical problems, the present invention provides a method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model, which specifically includes the following steps:

[0006] S1: 3D modeling of the spider aerial work vehicle;

[0007] S2: Import the 3D model obtained in step S1 into ADAMS software, perform Boolean operations on the model to merge it, and add the materials of each component, as well as the constraint pairs, moving pairs, contact forces, loads and drives between each part;

[0008] S3: In ANSYS software, each section of the vehicle's boom is flexibly processed to form a flexible body, and a modal neutral file is generated for export;

[0009] S4: Import the modal neutral file generated in step S3 into the ADAMS software and replace the rigid arm of the built model with a flexible body;

[0010] S5: Set up dynamic simulation in ADAMS software, including simulation duration and number of steps, and perform rigid-flexible coupling dynamic simulation;

[0011] S6: Use post-processing functions to obtain the stress and strain conditions at key hinge points, as well as the real-time changes in the contact reaction force between the outrigger and the ground;

[0012] S7: Based on the stress and strain obtained in step S6 and the real-time change of the contact reaction force between the legs and the ground, the overturning moment M of the spider aerial work vehicle is calculated using formula 1. r And use formula 2 to calculate the stability moment M of the spider aerial work vehicle s ;

[0013] S8: Combine the overturning moment M calculated in step S7 r and stabilizing torque M s , use formula 3, formula 4 and formula 5 to calculate the safety factor K.

[0014] Furthermore, the specific operations of adding constraint pairs and moving pairs in step S2 include: adding the center of mass of each component; adding a fixed pair between the support leg and the frame; adding a rotation pair between the center of rotation and the frame, adding a rotation pair between the first-level arm and the center of rotation, adding a rotation pair between the connecting frame and the first-level arm, adding a rotation pair between the connecting frame and the second-level arm, adding a rotation pair between the second-level three-section arm and the crank arm, adding a rotation pair between the crank arm and the bracket, adding a rotation pair between the bracket and the work platform; adding a moving pair between the second-level one-section arm and the second-level two-section arm, and adding a moving pair between the second-level two-section arm and the second-level three-section arm.

[0015] Furthermore, the contact force in step S2 includes: rigid-body-rigid-body contact force between the four legs and the ground.

[0016] Furthermore, the drive in step S2 includes: rotation drive between the rotation center and the frame, rotation drive between the bracket and the work platform; translation drive between the cylinder of the first-level arm hydraulic cylinder and the push rod, translation drive between the second-level arm hydraulic cylinder and the push rod, translation drive between the second-level one-section arm and the second-level two-section arm, translation drive between the second-level two-section arm and the second-level three-section arm, translation drive between the crank arm hydraulic cylinder and the push rod, and translation drive between the bracket cylinder and the push rod.

[0017] Furthermore, the load acting position in step S2 is a fixed load pointing vertically downward at the center of the working platform.

[0018] Furthermore, the specific operations of step S3 are as follows:

[0019] S3.1: Import the 3D model into the ANSYS APDL work interface;

[0020] S3.2: Set material properties and create external connection points;

[0021] S3.3: Use the Mesh module to perform meshing;

[0022] S3.4: Establish rigid regions, including establishing master and slave nodes;

[0023] S3.5: Output modal neutral file. Set the analysis type and the number of extracted modes when outputting the file.

[0024] Furthermore, the specific operations of step S4 are as follows:

[0025] S4.1: Use the flexible body replacement rigid body function in ADAMS software to import the modal neutral file into the model;

[0026] S4.2: Align the center of mass of the flexible body with the center of mass of the part to be replaced;

[0027] S4.3: Re-add constraint pairs and drives;

[0028] S4.4: Select the STEP function as the driving function to be used.

[0029] Furthermore, the formula 1 in step S7 is:

[0030] Among them, f r It is the magnitude of the force that causes the vehicle to overturn; r The distance between the force causing overturning and the overturning line of the vehicle;

[0031] Formula 2 in step S7 is:

[0032] Among them, f s Is the magnitude of the force that maintains the stability of the vehicle; s It is the distance between the force that maintains stability and the overturning line of the vehicle.

[0033] Furthermore, the formula three in step S8 is: ∑M=0;

[0034] The formula 4 in step S8 is: r +M z +M g =M s ;

[0035] Formula 5 in step S8 is:

[0036] Among them, M s is the stabilizing torque; M z M is the moment generated by the outrigger reaction force; g is the moment of inertia, M r is the overturning moment.

[0037] After adopting the above design, the beneficial effects of the present invention are as follows:

[0038] 1. Using virtual prototype design methods can get rid of excessive dependence on physical prototypes, reduce R&D costs, shorten cycles, and improve product quality.

[0039] 2. It overcomes many drawbacks of traditional R&D, supports all-round testing, analysis and evaluation of products, and emphasizes virtual collaborative design in different fields, which is an innovation in the R&D model.

[0040] 3. By adjusting the parameters, the stress and deformation of the spider aerial work vehicle under different working conditions can be simulated, which can provide a more intuitive response to the stability of the vehicle.

[0041] 4. The present invention can optimize and improve the structure of the spider-type aerial work vehicle based on the simulation analysis results, with lower cost and higher efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0043] Figure 1 This is a flow chart of a method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model of the present invention;

[0044] Figure 2 This is a top view of the spider-type aerial work vehicle safety factor calculation method based on the rigid-flexible coupling model of the present invention;

[0045] Figure 3 It is a three-dimensional rigid model diagram in the method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model of the present invention;

[0046] Figure 4 This is an exploded view of the secondary telescopic arm in the safety factor calculation method of the spider-type aerial work vehicle based on the rigid-flexible coupling model of the present invention;

[0047] Figure 5 This is a rigid-flexible coupling flow chart of a spider-type aerial work vehicle in a method for calculating a safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model of the present invention;

[0048] Figure 6It is a schematic diagram of arm flexibility in the method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model of the present invention;

[0049] Figure 7 Schematic diagram of the change of the safety factor K in the safety factor calculation method of the spider-type aerial work vehicle based on the rigid-flexible coupling model of the present invention;

[0050] Explanation of the accompanying reference numerals: 1. Support leg; 2. Frame; 3. Rotation center; 4. First-stage arm cylinder; 5. First-stage arm; 6. Connecting frame; 7. Second-stage arm cylinder; 8. Second-stage one-section arm; 9. Second-stage two-section arm; 10. Second-stage two-section arm; 11. Crank arm cylinder; 12. Crank arm; 13. Bracket; 14. Bracket cylinder; 15. Working platform. DETAILED DESCRIPTION

[0051] The technical solution of the present invention will be described clearly and completely below. Obviously, the embodiments described are only some embodiments of the present invention, not all embodiments. All other embodiments derived by persons of ordinary skill in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0052] In the description of the present invention, it should be noted that certain words indicating orientation or positional relationships are only for the purpose of facilitating the description of the present invention and simplifying the description, and are not intended to indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation. Therefore, they should not be understood as limitations on the present invention.

[0053] In the description of the present invention, it should be noted that the term "connection" should be understood in a broad sense. For example, it can mean a fixed connection, a detachable connection, or an integral connection; it can mean a mechanical connection or an electrical connection; it can mean a direct connection or an indirect connection through an intermediate medium; and it can mean internal communication between two components. Those skilled in the art will understand the specific meanings of the above terms in the present invention in specific circumstances.

[0054] The present invention is further described in detail below through specific examples.

[0055] like Figure 1 The present invention provides a method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model, which specifically includes the following steps:

[0056] S1: 3D modeling of spider aerial work platform components in the SolidWorks software environment;

[0057] S2: Import the 3D model created in step S1 into ADAMS software, perform Boolean operations on the model to merge some parts into one component to facilitate subsequent simulation, and add the materials of each component (all added materials are steel), as well as the constraint pairs, moving pairs, contact forces, loads and drives between each part;

[0058] S3: In ANSYS software, each section of the vehicle's boom is made flexible to form a flexible body, and a modal neutral file (mnf) is generated for export;

[0059] S4: Import the modal neutral file (mnf) generated in step S3 into the ADAMS software and replace the rigid arm of the built model with a flexible body;

[0060] S5: Set up dynamic simulation in ADAMS software, including simulation duration and number of steps, and perform rigid-flexible coupling dynamic simulation;

[0061] S6: Use post-processing to obtain the stress and strain at key hinge points, as well as the real-time changes in the contact reaction force between the outrigger and the ground. Verify the accuracy of the model by detecting the stress changes at key hinge points.

[0062] S7: Based on the stress and strain obtained in step S6 and the real-time change of the contact reaction force between the legs and the ground, the overturning moment M of the spider aerial work vehicle is calculated using formula 1. r And use formula 2 to calculate the stability moment M of the spider aerial work vehicle s ;

[0063] S8: Combine the overturning moment M calculated in step S7 r and stabilizing torque M s , use formula 3, formula 4 and formula 5 to calculate the safety factor K.

[0064] The present invention utilizes a virtual prototype design method to eliminate over-reliance on physical prototypes, thereby reducing R&D costs, shortening development cycles, and improving product quality. It overcomes many of the drawbacks of traditional R&D, supports comprehensive product testing, analysis, and evaluation, and emphasizes virtual collaborative design across diverse fields, representing an innovation in R&D models. By adjusting parameters, the present invention can simulate the stress and deformation of spider-type aerial work platforms under different operating conditions, providing a more intuitive understanding of the vehicle's stability. Furthermore, the present invention can optimize and improve the spider-type aerial work platform's structure based on simulation analysis results, resulting in lower costs and higher efficiency.

[0065] like Figure 5The figure shows the operation flow of the rigid-flexible coupling model in the present invention, and the specific steps are as follows: the graphic file of the established three-dimensional rigid model is imported into the dynamics simulation software, and a redundancy check is performed through the software. If the redundancy check fails, it is returned to the dynamics simulation software. Otherwise, the rigid-flexible coupling model is generated in combination with the finite element analysis software, and the support reaction force of the outrigger is obtained through the rigid-flexible coupling model.

[0066] like Figure 2 、 Figure 3 and Figure 4 The three-dimensional modeling of the spider-type aerial work vehicle components shown specifically includes a frame 2, a support leg 1 arranged at the bottom of the frame, a rotation center 3 arranged at the top of the frame, a first-level arm 5 connected to the rotation center, a first-level arm cylinder 4, a second-level arm, a second-level arm cylinder 7, a crank arm 12, a crank arm cylinder 11, a bracket 13, a bracket cylinder 14 and a working platform 15 arranged between the first-level arm and the rotation center. The second-level arm is composed of a second-level one-section arm 8, a second-level two-section arm 9 and a second-level three-section arm 10 connected in sequence. The second-level one-section arm is connected to the second-level one-section arm through a connecting frame 6. The second-level arm cylinder is arranged between the connecting frame and the second-level one-section arm. One end of the crank arm is connected to the second-level two-section arm through a mounting frame. The crank arm cylinder is arranged between the mounting frame and the crank arm. The working platform is connected to the other end of the crank arm through a bracket. The bracket cylinder is arranged between the bracket and the crank arm. The spider-type aerial work vehicle has four overturning lines AB, BC, CD, and DA.

[0067] In the above step S2, for the 3D model imported into ADAMS, six generalized coordinate Lagrangian equations with multipliers and corresponding constraint equations are established for each component according to the Lagrangian equation of motion:

[0068]

[0069] ψ i =0(i=1,2,3…,n);

[0070] Where: K is the expression of system kinetic energy; q j Describe the generalized coordinates of the system; ψ i The constraint equations of the system; F j Generalized force in the generalized coordinate direction; λ i m*1 Lagrange multiplier matrix.

[0071] The operations of adding constraint pairs in the above step S2 include: adding the center of mass of each component; adding a fixed pair between the support leg and the frame; adding a rotation pair between the center of rotation and the frame, adding a rotation pair between the first-level arm and the center of rotation, adding a rotation pair between the connecting frame and the first-level arm, adding a rotation pair between the connecting frame and the second-level arm, adding a rotation pair between the second-level three-section arm and the crank arm, adding a rotation pair between the crank arm and the bracket, adding a rotation pair between the bracket and the work platform; adding a moving pair between the second-level one-section arm and the second-level two-section arm, and adding a moving pair between the second-level two-section arm and the second-level three-section arm.

[0072] The contact force in step S2 includes: rigid-body-rigid-body contact force between the four legs and the ground.

[0073] The load in step S2 includes: simulating the weight of two workers and equipment on the working platform, setting the load to 200 kg, and the point of action at the center of the working platform.

[0074] The drive in the above step S2 includes: rotation drive between the rotation center and the frame, rotation drive between the bracket and the work platform; translation drive between the cylinder and the push rod of the first-level arm hydraulic cylinder, translation drive between the second-level arm hydraulic cylinder and the push rod, translation drive between the second-level first-section arm and the second-level second-section arm, translation drive between the second-level second-section arm and the second-level third-section arm, translation drive between the crank arm hydraulic cylinder and the push rod, translation drive between the bracket cylinder and the push rod; all use the STEP function in the built-in function formula in the ADAMS software as the driving function.

[0075] The driving functions used are all STEP functions, and the function format is: STEP(x,x0,h0,x1,h1);

[0076] Parameter Description:

[0077] x - independent variable, which can be time or any function of time

[0078] x0 - the starting value of the STEP function of the independent variable;

[0079] x1 – the end value of the STEP function of the independent variable;

[0080] h0 - initial value of the STEP function;

[0081] h1 - the final value of the STEP function.

[0082] The specific operations of the above step S3 are as follows:

[0083] S3.1: Import the 3D model into the ANSYS APDL work interface;

[0084] S3.2: Set material properties and create external connection points;

[0085] S3.3: Use the Mesh module to perform meshing;

[0086] S3.4: Establish rigid regions, including establishing master and slave nodes;

[0087] S3.5: Output modal neutral file. Set the analysis type and the number of extracted modes when outputting the file.

[0088] like Figure 6 As shown in the figure, after the arm frames of the vehicle are made flexible in ANSYS software, the deformation of the crank arm gradually increases from left to right.

[0089] Schematic diagram of arm flexibility in the method for calculating the safety factor of a spider-type aerial work vehicle using a rigid-flexible coupling model. The specific operation of the above step S4 is as follows:

[0090] S4.1: Use the flexible body replacement rigid body function in ADAMS software to import the modal neutral file into the model;

[0091] S4.2: Align the center of mass of the flexible body with the center of mass of the part to be replaced;

[0092] S4.3: Re-add constraint pairs and drives;

[0093] S4.4: Select the STEP function as the driving function to be used.

[0094] In the simulation of step S5 above, the simulation time is set to 200s, the number of simulation steps is set to 2000 steps, the turntable 3 rotates to the specified working position, and the primary arm 5 moves up and down under the action of the variable amplitude cylinder 4. After the primary arm 5 reaches the corresponding position, the secondary arm 8 begins to extend and retract. The secondary arm includes three telescopic arms. After completing the lifting action, each telescopic arm extends in turn, and finally the crank arm 12 performs the amplitude change. The main function of the crank arm is to adjust the working platform to the required working position.

[0095] The formula 1 in step S7 above is:

[0096] Among them, f r It is the magnitude of the force that causes the vehicle to overturn; r The distance between the force causing overturning and the overturning line of the vehicle;

[0097] Formula 2 in step S7 is:

[0098] Among them, f s Is the magnitude of the force that maintains the stability of the vehicle; s It is the distance between the force that maintains stability and the overturning line of the vehicle.

[0099] In the above step S8, for any overturning line, there exists a formula three: ∑M=0;

[0100] And formula 4 is: M r +M z +M g =M s ;

[0101] Among them, M s is the stabilizing torque; M z M is the moment generated by the outrigger reaction force; g is the moment of inertia, M r is the overturning moment.

[0102] Formula 5, the safety factor, can be obtained by using formula 3 and formula 4.

[0103] like Figure 7 The spider-type aerial work vehicle shown in the figure proposes a safety factor K to judge the stability. When K is less than or equal to 1, the whole machine overturns. When K is greater than 1, the whole machine remains stable. The present invention can ensure the stability of the whole machine by adopting the above method. Its safety factor K is always greater than 1 as the simulation time increases.

[0104] The above are merely preferred embodiments of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions based on the principles of the present invention are within the scope of protection of the present invention. It should be noted that for those skilled in the art, various improvements and modifications that do not depart from the principles of the present invention should be considered within the scope of protection of the present invention.

Claims

1. A method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model, characterized by: The specific steps include: S1: 3D modeling of the spider aerial work vehicle; S2: Import the 3D model obtained in step S1 into ADAMS software, perform Boolean operations on the model to merge it, and add the materials of each component, as well as the constraint pairs, moving pairs, contact forces, loads and drives between each part; S3: In ANSYS software, each section of the vehicle's boom is flexibly processed to form a flexible body, and a modal neutral file is generated for export; S4: Import the modal neutral file generated in step S3 into ADAMS software, and replace the rigid arm of the established model with a flexible body to establish a rigid-flexible coupling model; S5: Set up dynamic simulation in ADAMS software, including simulation duration and number of steps, and perform rigid-flexible coupling dynamic simulation; S6: Use post-processing functions to obtain the stress and strain conditions at key hinge points, as well as the real-time changes in the contact reaction force between the outrigger and the ground; S7: Based on the stress and strain obtained in step S6 and the real-time change of the contact reaction force between the legs and the ground, the overturning moment M of the spider aerial work vehicle is calculated using formula 1. r And use formula 2 to calculate the stability moment M of the spider aerial work vehicle s ; S8: Combine the overturning moment M calculated in step S7 r and stabilizing torque M s , use formula 3, formula 4 and formula 5 to calculate the safety factor K.

2. The method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model according to claim 1 is characterized in that: The specific operations of adding constraint pairs and moving pairs in step S2 include: adding the center of mass of each component; adding a fixed pair between the support leg and the frame; adding a rotation pair between the center of rotation and the frame, adding a rotation pair between the first-level arm and the center of rotation, adding a rotation pair between the connecting frame and the first-level arm, adding a rotation pair between the connecting frame and the second-level arm, adding a rotation pair between the second-level three-section arm and the crank arm, adding a rotation pair between the crank arm and the bracket, adding a rotation pair between the bracket and the work platform; adding a moving pair between the second-level one-section arm and the second-level two-section arm, and adding a moving pair between the second-level two-section arm and the second-level three-section arm.

3. The method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model according to claim 1 is characterized in that: The contact force in step S2 includes: rigid-body-rigid-body contact force between the four legs and the ground.

4. The method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model according to claim 1 is characterized in that: The driving in step S2 includes: rotational drive between the rotation center and the frame, rotational drive between the bracket and the working platform; translational drive between the hydraulic cylinder of the first-level arm and the push rod, translational drive between the hydraulic cylinder of the second-level arm and the push rod, translational drive between the second-level first-section arm and the second-level second-section arm, translational drive between the second-level second-section arm and the second-level third-section arm, translational drive between the crank arm hydraulic cylinder and the push rod, and translational drive between the bracket cylinder and the push rod.

5. The method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model according to claim 1 is characterized in that: The load in step S2 acts on a fixed load vertically downward from the center of the working platform.

6. The method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model according to claim 1 is characterized in that: The specific operations of step S3 are as follows: S3.1: Import the 3D model into the ANSYS APDL work interface; S3.2: Set material properties and create external connection points; S3.3: Use the Mesh module to perform meshing; S3.4: Establish rigid regions, including establishing master and slave nodes; S3.5: Output modal neutral file. Set the analysis type and the number of extracted modes when outputting the file.

7. The method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model according to claim 1 is characterized in that: The specific operations of step S4 are as follows: S4.1: Use the flexible body replacement rigid body function in ADAMS software to import the modal neutral file into the model; S4.2: Align the center of mass of the flexible body with the center of mass of the part to be replaced; S4.3: Re-add constraint pairs and drives; S4.4: Select the STEP function as the driving function to be used.

8. The method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model according to claim 1 is characterized in that: Formula 1 in step S7 is: Among them, f r It is the magnitude of the force that causes the vehicle to overturn; r The distance between the force causing overturning and the overturning line of the vehicle; Formula 2 in step S7 is: Among them, f s Is the magnitude of the force that maintains the stability of the vehicle; s It is the distance between the force that maintains stability and the overturning line of the vehicle.

9. The method for calculating the safety factor of a spider-type aerial work vehicle based on a rigid-flexible coupling model according to claim 1 is characterized in that: Formula 3 in step S8 is: ∑M=0; The formula 4 in step S8 is: M r +M z +M g =M s ; Formula 5 in step S8 is: Among them, M s is the stabilizing torque; M z M is the moment generated by the outrigger reaction force; g is the moment of inertia, M r is the overturning moment.

Images