Basic Vibration Problems
1. Vibration Forward Problem
Given the excitation (dynamic load) and structural parameters, solve for the vibration response of the structure.
This is known as the vibration forward problem, where the dynamic response of the structure is derived based on structural dynamics analysis theory.
2. First-Type Inverse Vibration Problem
Given the excitation and vibration response, determine the structural parameters.
This is known as the first-type inverse vibration problem, or system identification problem.
3. Second-Type Inverse Vibration Problem
Given the structural parameters and vibration response, determine the excitation.
This is known as the second-type inverse vibration problem—(dynamic) load identification problem.
Vibration System Models
1. Physical Parameter Model
A model characterized by mass, stiffness, and damping as the main parameters.
2. Modal Parameter Model
- One type: characterized by modal frequency, modal shape, and damping coefficient.
- Another type: characterized by modal mass, modal stiffness, modal damping, and modal vectors (residues).
3. Non-Parameter Model
Frequency response function (transfer function) or impulse response function can reflect the characteristics of a vibrating structure, referred to as non-parameter models.
Relationship
These three models are equivalent:
- From the physical parameter model (mass, stiffness, damping), the modal parameter model (modal frequencies, damping coefficients, etc.) can be derived, and then the non-parameter model (frequency response or impulse response function) can be obtained.
These are the fundamental concepts of vibration theory and form the theoretical basis for system identification.
System Identification of Vibration Structures
1. Physical Parameter Identification
Based on the physical model of the structure, the physical parameters are the target of identification. This is the basis for structural dynamic modification.
2. Modal Parameter Identification
- Based on the modal parameter model, modal parameters are the target of identification.
- Advantage: Modal parameters reflect the inherent vibration characteristics of the structure overall, and fewer parameters need to be identified.
- Modal parameter identification is the fundamental requirement of system identification and is the basis of physical parameter identification, being the main task of modal analysis.
3. Non-Parameter Identification
Determine the frequency response function (or transfer function) or impulse response function of the structure based on the excitation and response, reflecting the structural characteristics.
Modal Analysis Concepts
Narrow Definition
Modal analysis is a method based on structural vibration theory with the aim of identifying modal parameters.
Broad Definition
Modal analysis studies the relationships between the physical parameter model, modal parameter model, and non-parameter model of a structure, and determines the theoretical and application models of these systems through certain methods.
Two Modal Analysis Processes
Modal analysis is divided into theoretical modal analysis and experimental modal analysis based on specific methods and techniques.
1. Theoretical Modal Analysis
Theoretical modal analysis is based on linear vibration theory and studies the relationship between excitation, structure, and response. The modal parameter model is derived from the physical parameter model of the structure, and then the non-parameter model is obtained.
2. Experimental Modal Analysis
Experimental modal analysis is the experimental process of modal analysis and is the reverse of theoretical modal analysis:
- First, conduct vibration experiments on the structure to measure the time histories of excitation and response.
- Use signal processing techniques to determine the frequency response function (transfer function) or impulse response function, thus obtaining the non-parameter model.
- Then, use parameter identification methods to determine the system’s modal parameters, and if needed, further derive the physical parameters of the structure.
Experimental modal analysis integrates linear vibration theory, dynamic testing principles and methods, digital signal processing, and parameter identification techniques to identify structural parameters.
What is Modal Parameter Identification and Its Classification Methods
Concept
Modal parameter identification is based on the modal parameter model, with modal parameters as the target of identification.
Modal parameters reflect the inherent vibration characteristics of the structure, and fewer parameters need to be identified. Therefore, modal parameter identification is the basic requirement for system identification and is the foundation of physical parameter identification.
Classification Methods
1. Based on the Parameter Model
- Frequency domain parameter identification
- Time domain parameter identification
2. Based on the Number of Response Signals
- Local identification
- Global identification
3. Based on the Number of Exciation and Response Signals
- SISO Identification (Single Input, Single Output) — Local identification.
- SIMO Identification (Single Input, Multiple Outputs)
- MIMO Identification (Multiple Inputs, Multiple Outputs) — Global identification.
- In SISO identification, further classified into single-modal identification and multi-modal identification depending on the modal density of the structure.
4. Based on Modal Parameter Identification Methods
- Graphical identification method
- Resonance peak method
- Component analysis method
- Vector diagram analysis method
- Computer-based identification method
5. Based on Development Stages
- SISO Identification (Early 1970s)
- SIMO Identification (Late 1970s)
- MIMO Identification (1980s)
Damping Models in Modal Analysis
Common damping models include the following:
- Viscous Proportional Damping (Linear damping model)
- General Viscous Damping
- Structural Proportional Damping (Linear damping)
- Structural Damping
Real Modal Analysis
In undamped and proportional damping systems, the modal vectors are real, and these structures are called real modal systems.
Complex Modal Analysis
For structural damping and general viscous damping systems, the modal vectors are complex, and these structures are called complex modal systems.