Transient Structural Analysis
Transient structural analysis (also known as dynamic analysis) is a method used to determine the dynamic response of a structure over time. Through the analysis of transient analysis, we can obtain the time-dependent results such as displacement, strain, stress, and reaction force of the structure under arbitrary combinations of steady-state loads, transient loads, and harmonic loads. The biggest advantage of transient analysis over static analysis is that it considers inertia and damping effects.
Common analysis methods for transient problems:
- Time-domain transient analysis
- Eigenvalue extraction (natural frequency and modal)
- Steady-state response (harmonic response analysis in the frequency domain)
- Spectrum response analysis (peak response calculation of shock)
- Random response analysis (vibration caused by random excitation)
Because the time-domain transient method is the most intuitive, it can solve a variety of linear and nonlinear problems, and has often been applied in the industry. This article only introduces the time-domain transient method. Other methods will be introduced in the future.
Governing Equation
The fundamental equation of motion for transient dynamics is:
Where [M] is the mass matrix. [C] is the damping matrix. [K] is the stiffness matrix. {u_tt} is the nodal acceleration. {u_t} is the nodal speed. {u} is the nodal displacement. {F} is the load. It can be seen that the transient structural governing equation is an equation containing second-order time derivatives. There are many methods for solving the second-order time derivative. The most widely used in structural finite elements is the Newmark implicit time integration method. WELSIM’s default time solver for structural analysis is also the Newmark method.
Common Solving Methods
There are three common finite element methods for transient dynamics: full method, reduced method, and modal superposition method.
- Full Method: Uses complete system matrices to calculate the dynamic response. It supports solving various nonlinear characteristics such as plasticity, large deformation, large strain, etc. The main disadvantage is that it is expensive and time-consuming.
- Reduced Method: Compresses data size using principal degrees of freedom and reduced matrices. It is faster and less expensive than the full method but has limitations in terms of applying boundary conditions.
- Modal Superposition Method: Calculates the structural response by multiplying the mode shapes obtained by modal analysis. It is faster but does not support non-linearities except for node-to-node contact.
Damping
Considering the effect of damping is one of the advantages of transient analysis. Damping can be regarded as a type of energy dissipation. There are three common damping settings in FEM: direct damping, Rayleigh damping, and composite damping.
In most linear dynamic problems, accurately defining damping is crucial for results.
Time Solver
For transient structural problems, we need a time solver.
We generally divide time solvers into two categories: explicit solvers and implicit solvers.
- Explicit Solvers: Calculate the next result using the previous and current steps. Requires small time steps.
- Implicit Solvers: Iterates the next step result with current results, allowing larger time steps but requiring more computational resources.
Boundary and Initial Conditions
Transient analysis supports velocity and acceleration boundary conditions. The initial conditions it generally supports include:
- Linear static results.
- A transient analysis result at a certain moment.
- Initial velocity and initial acceleration of all free nodes.
Structural Transient Analysis Steps
Create or Import a Model
Meshing
Load Step and Time Step Settings
Set Boundary Conditions, Initial Conditions, Contacts
Solve and Verify Results
It can be seen that no damping occurs in this analysis because of no attenuation of the reciprocating vibration. The role and setting of damping will be introduced in the future.