Analysis of calculation characteristics and examples of force me
1. Gravity g = mg
(vertically downward, g = 9.8m/s2 ≈ 10m / S2, the action point is at the center of gravity, applicable to the earth surface)
2. Hooke's law f = KX
{along the recovery deformation direction, K: stiffness coefficient (n / M), X: deformation variable (m)}
3. Sliding friction force F = μ FN
{opposite to the relative motion direction of the object, μ： Friction coefficient, FN: positive pressure (n)}
{rrrrrrr}
extended data:
different classification of force
1. According to the nature of force, it can be divided into gravity, universal gravitation, elastic force, friction force, molecular force, electromagnetic force, nuclear force, etc Note that gravity is not equal to gravity under all conditions Gravity does not point to the center of the earth under all conditions. Gravity is a component of the earth's gravitational force on an object, and the other component is a centripetal force. Only on the equator does gravity point to the center of the earth.)
According to the effect of force, it can be divided into tension, tension, pressure, supporting force, power, resistance, centripetal force, restoring force, etc According to the research object, it can be divided into external force and internal force According to the action mode of force, it can be divided into non-contact force (such as gravitation, electromagnetic force, etc.) and contact force (such as elastic force, friction force, etc.) There are four basic interactions (forces): gravitational interaction, electromagnetic interaction, strong interaction and weak interactionnature of force:
materiality: force is the effect of an object (matter, mass) on an object (matter, mass). When an object is subjected to a force, another object must exert this effect on it. Force cannot exist independently without an object
interactivity (interaction): the interaction between any two objects is always mutual, and the object exerting the force must also be the object under the force. As long as one body exerts a force on another, the stressed body in turn will surely add a force to the exerted body Generating conditions: the force is equal in size (the resultant force is zero, in a state of non directional static motion) or not equal, in the opposite direction, acting on two different objects, and acting on the same straight line. It can be summarized as: foreign body, equivalent, reverse, collinear. A pair of interaction forces must proce and disappear at the same time.)
Vectoriality: force is a vector, which has both magnitude and direction
simultaneity: the force proced and disappeared at the same time
independence: the effect of one force does not affect the effect of another
includes three elements: the size, direction and action point of the force. The accurate expression of the three elements of force by a directed line segment is called the diagram of force. The size is represented by the length of a scaled line segment, the direction is represented by an arrow, the point of action is represented by an arrow or the tail of an arrow, and the straight line along which the direction of a force follows is called the line of action of a force. The diagram of the force is used for the calculation of the force. When the judgment power is large, we must pay attention to the scale of the line segment, because even if one line segment is longer than another line segment, but the scale of the long line segment is also longer, the force represented by the short line segment is not necessarily smaller than that represented by the long line segment
exercise example 1
quick calculation method exercise example
example of rapid calculation in practice
> Shi Fengshou calculation method is easy to learn and use. The algorithm starts from high order numbers, and memorizes 26 pithy formulas summarized by Professor Shi (these pithy formulas do not need to be memorized, but conform to scientific laws and are connected with each other), It is used to express the carry rule of one digit multiplied by many digits. If you master these pithy formulas and some specific rules, you can quickly carry out operations such as addition, subtraction, multiplication, division, power, square root, fraction, function, logarithm, etc
? This paper gives an example of multiplication,
ofast algorithm, like traditional multiplication, needs to process each digit of the multiplier bit by bit. We call the digit being processed in the multiplier "standard", and the number from the first digit to the last digit on the right side of the standard "post digit". After the standard is multiplied, only the single digit of the proct is taken, which is called "Ben Ge", and the number to be carried after the standard's last digit is multiplied by the multiplier is called "backward"
oeach digit of the proct is the digit of the sum of "this proct plus the last", i.e. ---
- the digit of the sum of the standard proct = (this proct plus the last ten)
othen we need to calculate this proct and the last one bit by bit from left to right, and then add them together to get the single digit. Now, take the right example to illustrate the thinking activities in calculus
(example) add 0 to the first place of the multiplicand, and list the formula:
0847536 × 2 = 1695072
the carry rule of multiplier 2 is "2 full 5 into 1"
0 × Two zeros, eight in the back, one in the back, get 1
8 × If you don't enter, you get 6
4 × The second is 8, the second is 7, the full 5 enters 1,
8 + 1 gets 9
7 × 2 this 4, after 5, full 5 into 1,
4 ten 1 get 5
5 × Two zeros, the last three not into, get 0
3 × 2 this 6, after 6, full 5 into 1,
6 ten 1 get 7
6 × In this paper, we only give the simplest examples for readers' reference. As for multiplying by 3, 4... And 9, there are also some carry rules, which can not be listed one by one e to space limitation< Based on these carry rules, Shifeng harvest speed algorithm is developed step by step. As long as you are proficient in using four multi digit arithmetic operations, such as addition, subtraction, multiplication and division, you can achieve the purpose of fast and accurate.
The basic unknown quantity calculated by force method is the rendant binding force X
in statically indeterminate systems, all external constraints are involved, that is, rendant constraints; Rendant constraints are additional constraints on statically determinate structures. Each rendant constraint brings a rendant unknown generalized force, which makes the total number of generalized forces exceed the total number of independent equilibrium equations listed. The excess number is called the statically indeterminate degree or statically indeterminate degree of the structure
extended data:
the calculation steps of force method are summarized as follows:
(1) determine the statically indeterminate times of the original structure
(2) the statically determinate basic structure is selected (it is called the basic structure after removing the rendant constraints, and the equivalent system of the original structure is obtained by replacing the rendant constraints with the rendant unknown forces)
(3) write the typical equation of force method (4) the internal force diagram of each unit and the internal force diagram of load of the equivalent system are made, and the coefficients and free terms in the typical equation are calculated accordingly(5) typical equations are solved to find out the rendant unknown forces
(6) the internal force diagram was made by superposition (7) check. Static balance check + displacement condition check7.3.1 determination of proctivity equation × 10-3 μ M < sup > 2 < / sup >, the formation temperature is 392k, the formation pressure is 47.25mpa, the viscosity of natural gas is 0.0252mpa · s, the compression factor of natural gas is 0.92, the radius of gas reservoir is 1500m, the radius of well is 0.1M, the thickness of reservoir is 5.6m, the empirical constant is 6.22, and the relative density of natural gas is 0.72
After coring, the well was drilled to a depth of 3600 M. The corrected isochronal test data of the well are shown in Table 7.1
Table 7.1 modified isochronal test pressure and proction data of a well
Fig. 7.7 relationship between damage radius and skin
are very detailed
in displacement method analysis, the typical equation of displacement method listed is the balance equation. In the process of equation listing, the balance equation is listed with displacement as unknown quantity. By solving the typical equation of displacement method, the deformation coordination relationship is automatically satisfied, and the displacement method reflects the balance condition and deformation condition
in the same way, the solution of the force normal equation automatically satisfies the equilibrium condition.
1. Diagonal rule
this rule is applicable to calculate the value of low-order determinant (such as the value of second-order and third-order determinants), that is, the proct of the elements on the main diagonal minus the proct of the elements on the secondary or sub diagonal. Its main idea is to calculate the value of the determinant according to the definition of second-order and third-order determinants
2. Change the determinant into triangular determinant
by using the properties of determinant, change the determinant into upper (lower) triangular determinant, and then use the conclusion of upper (lower) triangular determinant, we can get the value of the corresponding determinant
upper (lower) triangular determinant and its value (1) the upper triangular determinant is d = | ■ (■ (a)_ 11&a_ 12@0_ & a_ 22 )&■(a_ 13&…&a_ 1n@a_ 23&…&a_ 2n )@■(0_ & 0_ @&# 8942;&&# 8942;@ 0_ & 0_ )& ■(a_ 33&…&a_ 3n@⋮&&# 8942;&&# 8942;@ 0_ &…& a_ nn ))|
D=|■(■(a_ 11&a_ 12@0_ & a_ 22 )&■(a_ 13&…&a_ 1n@a_ 23&…&a_ 2n )@■(0_ & 0_ @&# 8942;&&# 8942;@ 0_ & 0_ )& ■(a_ 33&…&a_ 3n@⋮&&# 8942;&&# 8942;@ 0_ &…& a_ nn ))| =|■(■(a_ 11&0&0@a_ 21&a_ 22&0@a_ 31&a_ 32&a_ 33 )&■(⋯& 0@⋯& 0@⋯& 0)@■(⋮&&# 8942;&&# 8942;@ a_ n1&a_ n2&a_ n3 )&■(⋮&&# 8942;@&# 8943;& a_ nn ))| = a_ 11 a_ 12⋯ a_ NN
that is, the value of the determinant of the upper (lower) triangle is equal to the proct of the elements on the main diagonal
the general steps of the triangulation method are as follows: (taking the upper triangular determinant as an example)
the first step is to put a_ Transform 11 to 1 or multiply the first line by 1 to achieve 1 / A_ 11 to achieve, try to avoid the score
Step 2: multiply the first line by - a respectively_ 21,-a_ 31，⋯,- a_ N1 is added to the corresponding elements in rows 2, 3,..., N, and the first column A is added_ All elements below 11 are reced to 0
Step 3: from the second line, turn the main diagonal a with the above method_ 22,a_ 33,⋯ a_( All the elements below (n-1, n-1) are reced to 0, which leads to the upper triangular determinant
3. Split method
the elements of a row (or column) are written in the form of the sum of two numbers, and then the original determinant is written in the form of the sum of two determinants by using the property of determinant, which simplifies the problem and facilitates the calculation< (1) recursive rection method: recursive rection method can be divided into direct recursion and indirect recursion. The key of calculating determinant with direct recursion method is to find out an algebraic expression to express the value which can be obtained by recursion step by step; The method of indirect recursion is to transform the original determinant to construct a system of equations about sum, and then the solution can be obtained by elimination
(2) according to the theorem expansion method: according to the determinant expansion theorem, the given determinant can be expanded into the sum of several lower order determinants. If the determinant can be transformed so that only one element of a row (column) is not zero, then the determinant can be transformed into a lower order determinant for calculation< In order to simplify the problem, we usually use the property of determinant to transform the given determinant, and then use the expansion theorem to rece the order. Sometimes, on the contrary, adding rows and columns on the basis of the original determinant, so that it can be upgraded to construct a new determinant which is easy to calculate, and then calculate the value of the original determinant. This method of calculating determinant is called order raising method. When upgrading, which elements make up the new row (column)? Where to add? The proper choice should be made according to the characteristics of the original determinant
6. Undetermined coefficient method
this method is an important method in mathematics. According to the inherent characteristics of solving the problem, it can be transformed into an identity with undetermined coefficient, and then the unknown coefficient can be obtained by using the property of the identity, so as to obtain the method of solving the problem. The idea of solving the determinant with undetermined coefficient method is: if there are indeterminate elements in the determinant, Then the determinant must be a polynomial about, and when some values are taken, such as making the value of determinant zero, according to the theory of polynomial integral division, the determinant must be divisible by this linear factor, that is, the expression of the determinant should contain this factor. If all the factors of the determinant can be found, the value of the determinant can be obtained by calculating the undetermined constant
7. Mathematical inction
that is to use incomplete inction to find the conjecture value of travel formula, and then use mathematical inction to give a strict proof of the conjecture value. The second type of mathematical inction is often used here
8. Proct method
in determinant, if every element can be decomposed into the sum of procts (a)_ i1 b_ 1j+a_ i2 b_ 2j+…+a_ in b_ NJ), then the determinant can be transformed into the determinant of the proct of two matrices. As long as the determinant of the decomposed two matrices is easy to calculate, it can be solved by the formula | ab | = | a | &# 8729| B | calculate the value of the original determinant
9. Bordering method
generally, when calculating the determinant, we rece the price, but for some determinants, we can conversely add one row one column (the added elements of one row one column are generally composed of 1 and 0) d on the basis of keeping the value of the original determinant unchanged_ n=|■(1&■(0&⋯& 0)@■(x_ 1@⋮@ x_ n )&D_ n )|_( n+1) or |■(1&■(x_ 1&⋯& x_ n )@■(0@⋮@ 0)&D_ n )|_( N + 1)
to transform n-order determinant into N + 1-order determinant, we only need to select x skillfully_ 1,x_ 2 ,…,x_ N, combined with the property of determinant, the value of determinant can be calculated
for the calculation of determinant, it is often difficult and easy e to different methods, and the difference is very large. In order to make the calculation process simple and clear, we should be good at choosing appropriate methods and master certain skills. To explore and summarize these skills is not only of practical significance, but also of profound theoretical significance. Therefore, I will focus on the use of various methods, various types of topics most suitable for which method. At present, the most commonly used methods are some more conventional methods, but often these methods have a large amount of calculation, so we are faced with the problem of how to promote the skills that we have not used frequently< References:
1 Zhang horui, Hao Bingxin. Higher algebra. Higher ecation press, 1988
2. Wang Yufang. Course of higher algebra. Tsinghua University Press, 1997
3. Yao mussen. Higher algebra. Fudan University Press, 2002
Dai Hua. Matrix theory. Science Press, 2001
Wang Zuozhong. Calculation methods and skills of determinant [J]. Private science and technology, issue 08, 2010
Han Baoyan. Calculation methods and application of determinant [J]. Science and technology information, issue 03, 2010
Chen Huiping. Research on calculation methods of n-order determinant [J]. Heilongjiang Science and technology information, issue 03, 2010
1. The statically indeterminate times of the original structure are determined
2. Select the statically determined basic structure (after removing the rendant constraints, it is called the basic structure, and the equivalent system of the original structure is obtained by replacing the rendant constraints with the rendant unknown forces)
3. Write the typical equation of force method The internal force diagram of each unit and the internal force diagram of load of the equivalent system are made, and the coefficients and free terms in the typical equation are calculated accordingly The typical equations are solved and the rendant unknown forces are obtained The internal force diagram was made by superposition7. Static balance check + displacement condition check
The common constraint types of stress analysis are as follows: (1) the constraint is placed along the normal line of the contact surface.(2) the constraint is perpendicular to the axis of rotation, but the direction is uncertain. It is usually expressed by two component forces which are perpendicular to each other and perpendicular to the axis of rotation.
(3) the constraint is over the center of the ball, (4) the binding force of the roller seat is perpendicular to the contact surface of the roller seat.
(5) the binding force of the journal bearing and the thrust bearing is perpendicular to the rotating shaft, but its direction is unknown, The thrust bearing is equal to the journal bearing, plus the applied force, three components can be drawn, one component is along the axis, and the other two components are perpendicular to each other
for the mechanical calculation of complex structure, sometimes it is necessary to separate each component from the joint and draw the force diagram of each component separately. At this time, it must be noted that the force diagram shows that the binding force at the joint obeys the law of force and reaction