Chapter 3 Virtual Displacement Principle 3

Example 15.6 known: As shown in truss structure 15.10 (a), the concentration force of the D node is calculated as the internal force of one or two poles.

Solution: 1. Calculate the internal force of 1 pole

Lift the constraint of 1 pole, force, and Main Power System

Any virtual displacement to the system, as shown in 15.10 (B.

Principle of Virtual Displacement:, ()

Virtual Displacement relationship = 6 (B)

Substitute formula (B) into formula (a),

Is independent. Here, sin is obtained.

2. Calculate the internal force of 2 rods.

Lift the pole constraint and apply force

Main power system ()

Any virtual displacement to the system, as shown in 15.10 (c.

Virtual Displacement principle:

Is independent.

6. Equilibrium Conditions of point systems expressed in Generalized Coordinates

1. General System

Set the function relationship between the bitvector and the generalized coordinate of the prime point in the prime point system

Let's get (a) The above variational formula)

Using formula (a) into the theory of virtual displacement (),

Exchange summation order, with (B)

Order (c) is called the generalized force corresponding to the generalized coordinate or

(D)

Type (B) becomes

For a complete constrained system, the variation of the generalized coordinates is a prime point system with complete, dual-sided, and stable ideal constraints. The necessary and sufficient conditions for maintaining a balance at a given position are, the generalized force corresponding to each generalized coordinate is equal to zero.

2. Conservative System

If the function corresponds to a powerful potential energy function

, (E)

Substitute formula (e) into formula (d),

Total potential energy functions of the system, including

The balance condition for a conservative system is

3. Methods for obtaining the advertising force

1) Resolution Method: by definition, that is

2) ry: the equilibrium condition expressed by the generalized force

That is

Ling

Ling

3) if the main power is powerful, write the potential function V and use it.

Please

For example, 15.7 is known: As shown in 15.11 (a), the OA and AB lengths of homogeneous rods are the same, the weight is the same, and the water flat force is applied to the B ends of the AB rod, and the balance is shown in the figure below. Q: What is the position of the two poles during balancing?

Solution: study object: Whole

Select the main dynamic system of generalized coordinates

1. Use the parsing method to solve the coordinates shown in 15.11 ().

;

Write the coordinate of each force vertex and obtain the partial derivative of the generalized coordinate.

Search for advertising Power

Order

2. Solving with ry

Order (Figure 15.11 (B ))

Virtual Displacement relationship:

Strive in a broad sense

Substituted into the virtual displacement relationship,

(Figure 15.11 (c ))

Virtual Displacement relationship:

Search for advertising Power

Substituted into the tiger displacement relationship,

Make available

3. All are powerful. The equipotential surface of the force is a direct plane.

The potential energy function of the system is

Search for advertising Power

Same as in the first two cases

Example 15.9 known: 15. As shown in 12, the pulley system is set to A, B, and C, and the weight is not counted.

When the system is balanced, the weight of weight C and the sliding friction coefficient between weight A and the horizontal plane.

Solution: study object: Whole

Select the generalized coordinates and the main power system.

Solving with ry

At this time

Guangyili

At this time

Guangyili

Order

Strive for friction coefficient, demand, lift horizontal constraints, add, order

And

Therefore, the friction coefficient between thing A and the platform is