What Are Equations of Motion?
The equation of motion is a mathematical expression that describes the relationship between force and displacement (including speed and acceleration) in a structure. There are five main methods for its establishment, including Newton's second law, D'Alembert's principle, virtual displacement principle, Hamilton's principle, and Lagrange's equation.
- The equation of motion is a mathematical expression that describes the relationship between force and displacement (including speed and acceleration) in a structure.
- single
- The establishment of the equation of motion mainly consists of the following five methods. Take the motion of the single particle system on the right as an example.
- Newton's second law
- According to the force analysis of the single particle system, the equation of motion of the single particle system can be directly written:
- Advantages of Newton's second law:
- Newton's second law is based on the direct application of existing knowledge in physics, and establishes the equation of motion of the system with the most easily accepted mechanical knowledge.
- D'Alembert principle (direct dynamic balance method)
- At any instant of system movement, if in addition to the main force (including damping force) and restraint reaction force of the actual acting structure, plus (imaginary) inertial force, the system will be in an assumed equilibrium state (dynamic balance).
- Advantages of the D'Alembert principle :
- Static problems are familiar to people. With the D'Alembert principle, the formal dynamic problems become static problems. The methods used to establish the governing equations in static problems can be used to establish the balance of dynamic problems The equation makes the thinking on the dynamic problem somewhat simplified. For many problems, the D'Alembert principle is the most direct and easiest way to establish the equation of motion, and establishes the concept of dynamic balance (abbreviated as: dynamic balance).
- Virtual displacement principle
- Principle of virtual displacement: When a virtual displacement occurs in a balanced system under the action of a group of external forces, the sum of the virtual work done by the external force on the virtual displacement is always equal to zero.
- Virtual displacements are infinitely small displacements that satisfy the constraints of the system. Suppose a virtual displacement du occurs in the system, and the total virtual work done by the equilibrium force on du is:
- among them,
- Advantages of the virtual displacement principle:
- The principle of virtual displacement is based on the analysis of virtual work, and virtual work is a scalar that can be calculated algebraically, so it is simpler than Newton's second law or the vector operation required in the D'Alembert principle.
- Hamilton principle
- Variational method (principle) can be used to establish the equation of motion of the structural system. Mathematically, the variational problem is the extreme value of a function. Here, a functional is energy (work) in a structural system.
- The equilibrium position of the system is the stable position of the system. At the stable position, the energy of the system obtains an extreme value, which is generally a minimum.
- Hamilton principle: In any time interval [t1, t2], the variation of the kinetic energy and the potential energy of the system plus the variation of the work performed by non-conservative forces is equal to zero.
- among them:
- Advantages of the Hamilton principle:
- It is not obvious to use inertial and elastic forces, but to use the variation of kinetic energy and potential energy respectively. So for these two, it only involves dealing with pure scalars, that is, energy. In virtual displacement, although the virtual work itself is a scalar, the force and virtual displacement used to calculate the virtual work are both vectors.
- Lagrange equation
- The Hamilton principle is a variational method for dynamic problems in integral form. In fact, there is another equivalent variational principle for dynamic problems in differential form, which is the Lagrange equation of motion. Its expression is as follows:
- among them:
- Advantages of Lagrange equation:
- Get more applications. It is the same as the Hamilton principle. It is a complete scalar analysis method except for conservative forces (damping forces). It is not necessary to directly analyze inertial and conservative forces (mainly elastic restoring forces). Restoring force is the most difficult object to deal with when establishing the equation of motion. [1]