STRUCTURAL DYNAMICS

STRUCTURAL DYNAMICS

--------------------------------------------------------------------------
1. Introduction to Structural Dynamics

2. Single Degree-of-Freedom Systems
    a. Fundamental Equation of Motion
    b. Free Vibration of Undamped Structures
    c. Free Vibration of Damped Structures
    d. Forced Response of an SDOF System

3. Multi-Degree-of-Freedom Systems
    a. General Case (based on 2DOF)
    b. Free-Undamped Vibration of 2DOF Systems

4. Continuous Structures
    a. Exact Analysis for Beams
    b. Approximate Analysis – Bolton’s Method

5. Practical Design
    a. Human Response to Dynamic Excitation
    b. Crowd/Pedestrian Dynamic Loading
    c. Damping in Structures
    d. Rules of Thumb for Design

6. Appendix 54
    a. References
    b. Important Formulae
    c. Important Tables and Figures
--------------------------------------------------------------------------

1. Introduction to Structural Dynamics

Modern structures are increasingly slender and have reduced redundant strength due to improved analysis and design methods. Such structures are increasingly responsive to the manner in which loading is applied with respect to time and hence the dynamic behaviour of such structures must be allowed for in design; as well as the usual static considerations. In this context then, the word dynamic simply means “changes with time”; be it force, deflection or any other form of load effect.

Examples of dynamics in structures are:
- Soldiers breaking step as they cross a bridge to prevent harmonic excitation;
- The Tacoma Narrows Bridge footage, failure caused by vortex shedding;
- the London Millennium Footbridge: lateral synchronise excitation.

 

(a)

                  Figure 1.1

(b)

The most basic dynamic system is the mass-spring system. An example is shown in Figure 1.1(a) along with the structural idealisation of it in Figure 1.1(b). This is known as a Single Degree-of-Freedom (SDOF) system as there is only one possible displacement: that of the mass in the vertical direction. SDOF systems are of great importance as they are relatively easily analysed mathematically, are easy to understand intuitively, and structures usually dealt with by Structural Engineers can be modelled approximately using an SDOF model (see Figure 1.2 for example).

                                                

                                           Figure 1.2       

If we consider a spring-mass system as shown in Figure 1.3 with the properties m = 10 kg and k = 100 N/m and if give the mass a deflection of 20 mm and then release it (i.e. set it in motion) we would observe the system oscillating as shown in Figure 1.3. From this figure we can identify that the time between the masses recurrence at a particular location is called the period of motion or oscillation or just the period, and we denote it T; it is the time taken for a single oscillation. The number of oscillations per second is called the frequency, denoted f, and is measured in Hertz (cycles per second).