A Comprehensive Guide to Vibration Analysis From Mathematical Modeling to Interpretation of Results

Vibration Analysis

The vibration analysis of a real system consists of the following steps:

1. Mathematical modeling of a real system: A mathematical model is used to determine the nature of the system, its features and aspects, and the components involved in the real system.

2. Formulation of governing equations: The differential equations governing the vibrating system’s behavior are derived using the laws of mechanics and dynamics after the mathematical model has been developed. Several concepts of the principles of dynamics such as Newton’s second law of motion, D’Alembert’s principle, and the principle of conservation of energy are most commonly used to formulate the governing equation of motion.

3. Solution of the governing equation of motion: The governing equations of motion of a vibrating system are solved to find the response of the vibration of an SDOF system leading to one ordinary differential equation of motion in the form of a second-order linear differential equation of motion. Vibrations of the MDOF system lead to a system of ordinary differential equations of motion. Similarly, the vibrations of continuous systems are governed by partial differential equations. There are many techniques available for solving the equation of motion, such as ordinary differential equations, matrix methods, finite element methods, and numerical methods.

4. Interpretation of results: The displacement, velocity, and acceleration of the various masses or inertia of the system are typically obtained by solving the equation of motion for the real physical system. The crucial and last phase in the various analysis processes is result interpretation. It includes drawings (which give the general inferences from results), development of design curves, and recommendations if any.

Mathematical Modelling of an SDOF System

To understand the dynamic behavior of the structures, it is necessary to develop their models under the influence of dynamic loads such as winds, blasts, earthquakes, heavy rotating machinery, etc. These models can be applied as mathematical models for research or analytical reasons, or they can be employed as laboratory models for conducting experimental studies.

Let us consider a simple portal frame. While developing a mathematical model, some assumptions are made to simplify the analysis. They are,

A Comprehensive Guide to Vibration Analysis From Mathematical Modeling to Interpretation of Results

(1) The total mass of a portal frame is assumed to act at the slab level since the masses of columns are much less when compared to that of the slab: i.e., masses of columns are ignored.

(ii) The beam/slab is assumed as infinitely rigid, so that the stiffness of the structure is provided only by columns, i.e., the flexibility of the slab/beam is ignored.

(iii) Since the beams are usually built monolithically within the columns, the beam-column joint can be assumed to be rigid without any rotations at the joint.

By these assumptions, the possibility of lateral deformation or displacement is due to only rigid beams/slabs. The model resulting from all the above-mentioned assumptions is called as shear building model. This shear-building idealization although unrealistic is necessary for the mathematical formulation of vibration problems.

The portal frame under the influence of a lateral load F(t) can be represented mathematically as the response of the SDOF system. This typical discrete spring-mass system is equivalent to the response of a portal frame. The parameters of the mathematical model are related to its prototype (actual frame):

1. Mass m represents the total mass of the beam and the slab of the frame and inertial characteristic of the structure; energy is stored by mass in the form of kinetic energy.

2. Stiffness of spring k represents the combined stiffness of two columns for lateral deformation that is the elastic restoring force and it stores the potential energy (internal strain energy) due to columns.

3. Dashpot having damping coefficient C represents the energy dissipation, i.e. the frictional characteristics and energy losses of the frame and

4. An excitation force F(r) represents the external lateral force applied on the portal frame.

Inactive or passive elements are the mass, the spring, and the damper or dashpot. Because these components contribute to vibratory motion yet are powerless to control vibrations. The function of the spring and the mass is to store energy while that of the damper is to dissipate the same in the form of a beat. The excitation element F(r) is called the active element through which energy is supplied to the vibratory system. It represents the source through which the energy flows into the physical system.

Such a model does not exist in the real world and mathematical models may provide complete and accurate knowledge of the dynamic behavior of the model itself. But in practice, the information acquired from the analysis of the model may be sufficient to study the dynamic behavior of the real system including design.

Since the essential properties of the dynamic system have been divided into independent discrete elements, such a model is also known as a lumped parameter model as against a distributed parameter model or continuous system wherein all properties are distributed continuously throughout.

From the above example, it is seen that the following four elements are the most important to determine dynamic behavior.