Chapter 1. The Need
Chapter 2. Basic Mechanics
Chapter 3. Response of a Linear One-Degree-of-Freedom Mechanical System to an Arbitrary Excitation
Chapter 4. Impulse and Step Responses
Chapter 5. Sinusoidal Vibration
Chapter 6. Response of a Linear One-Degree-of-Freedom Mechanical System to a Sinusoidal Excitation
Chapter 7. Non-Viscous Damping
Chapter 8. Swept Sine
Chapter 9. Response of a Linear One-Degree-of-Freedom System to a Swept Sine Vibration
Appendix: Laplace Transformations Vibration Tests: a Brief Historical Background
Chapter 2
Basic Mechanics
2.1. Basic principles of mechanics
2.1.1. Principle of causality
The state of the universe at a given moment determines its state at any later moment.
2.1.2. Concept of force
A force can be defined as any external cause able to modify the rest state or the movement of a material point.
A force is characterized by:
– its point of application (material point on which it acts); – its line of action, which is that of the straight line whereby it is applied; – its direction, which is that of the movement that it tends to produce; – its size (or intensity).
2.1.3. Newton’s first law (inertiaprinciple)
In the absence of any force, a material point, if it is at rest, remains at rest; if it is moving, it preserves a rectilinear and uniform motion.
2.1.4. Moment of a force around a point
Given a force and an arbitrary point O, the moment of the force around point O is defined as the product M = F d, where d is the perpendicular distance from point O to F (d is called the lever arm).
Figure 2.1 Lever arm for the calculation of the moment of a force
Let us set O' as the foot of the perpendicular to the support of drawn from O.
The moment M is positive if tends to make O' turn clockwise around O, negative in the contrary case.
2.1.4.1. Couple – moment of a couple
Two forces form a couple if they are parallel, of opposite direction and equal in size.
The moment of the couple is equal to M = Fd, where F is the amplitude of each force and d is the distance which separates them (couple lever arm).
2.1.5. Fundamental principle of dynamics (Newton’s second law)
The application of a force to the point of mass m involves a variation of its momentum, defined by the product of its mass by its instantaneous speed , according to the relation:
[2.1]
m is a coefficient characteristic of the body. If the mass m is invariable, the relation becomes:
[2.2]
i.e.
[2.3]
where is the acceleration undergone by the mass subjected to .
2.1.6. Equality of action and reaction (Newton’s third law)
If two particles isolated from the remainder of the universe are brought into each other’s presence, they exert upon each other two forces, carried by the line which joins them, of equal sizes and opposite directions. One is the action, the other the reaction.
2.2. Static effects/dynamic effects
In order to evaluate the mechanical characteristics of materials, it is important to be aware of the nature of stresses [HAU 65]. There are two main load types that need to be considered when doing this:
– those which can be considered as applied statically; – those which are the subject of a dynamic analysis of signal versus time.
Materials exhibit different behaviors under static and dynamic loads. Dynamic loads can be evaluated according to the following two criteria:
– the load varies quickly if it communicates significant velocities to the particles of the body in deformation, so that the total kinetic energy of the moving masses constitutes a large part of the total work of the external forces; this first criterion is that used during the analysis of the oscillations of elastic bodies; – the speed of variation of the load can be related to the velocity of evolution of the plastic deformation process occurring at a time of fast deformation whilst studying the mechanical properties of the material.
According to this last criterion, plastic deformation does not have time to be completed when the loading is fast. The material becomes more fragile as the deformation velocity grows; elongation at rupture decreases and the ultimate load increases (Figure 2.2).
Figure 2.2 Tension diagram for static and dynamic loads
Thus, a material can sometimes sustain an important dynamic load without damage, whereas the same load, statically, would lead to plastic deformation or failure. Many materials subjected to short duration loads have ultimate strengths higher than those observed when they are static [BLA 56], [CLA 49], [CLA 54], [TAY 46]. The important parameter is in fact the strain rate, defined by:
[2.4]
where is the deformation observed in time At on a test-bar of initial length subjected to stress.
If a test-bar of initial length 10 cm undergoes in 1 s a lengthening of 0.5 cm, the strain rate is equal to 0.05 s−1. The observed phenomena vary according to the value of this parameter. Table 2.1 shows the principal fields of study and the usable test facilities [AST 01], [DAV 04], [DIE 88], [LIN 71], [MEN 05], [SIE97]. This book will focus on the values in the region 10−1 to 101 s−1 (these ranges being very approximate).
Certain dynamic characteristics require the data of the dynamic loads to be specified (the order of application being particularly important). Dynamic fatigue strength at any time t depends, for example, on the properties inherent in the material concerning the characteristics of the load over time, and on the previous use of the part (which can reflect a certain combination of residual stresses or corrosion).
Table 2.1 Fields of strain rate
2.3. Behavior under dynamic load (impact)
Hopkinson [HOP 04] noted that copper and steel wire can withstand stresses that are higher than their static elastic limit and are well beyond the static ultimate limit without separating proportionality between the stresses and the strains. This is provided that the length of time during which the stress exceeds the yield stress is of the order of 10−3 seconds or less.
From tests carried out on steel (annealed steel with a low percentage of carbon) it was noted that the initiation of plastic deformation requires a definite time when stresses greater than the yield stress are applied [CLA 49]. It was observed that this time can vary between 5 ms (under a stress of approximately 352 MPa) and 6 s (with approximately 255 MPa; static yield stress being equal to 214 MPa). Other tests carried out on five other materials showed that this delay exists only for materials for which the curve of static stress deformation presents a definite yield stress, and the plastic deformation then occurs for the load period.
Under dynamic loading, an elastic strain is propagated in a material with a velocity corresponding to the sound velocity c0 in this material [CLA 54]. This velocity is a function of the modulus of elasticity, E, and of the density, ρ, of the material. For a long, narrow part, we have:
[2.5]
The longitudinal deflection produced in the part is given by:
[2.6]
where v1= velocity of the particles of the material. In the case of plastic deformation, we have [KAR 50]:
[2.7]
where is the slope of the stress deformation curve for a given value of the deformation ε. The velocity of propagation c is therefore a function of ε. The relation between the impact velocity and the maximum deformation produced is given by:
[2.8]
A velocity of impact v1 produces a maximum deformation ε1 that is propagated with low velocity since the deformation is small. This property makes it possible to determine the distribution of the deformations in a metal bar at a given time.
Most of the materials present a total ultimate elongation which is larger at impact than for static loading (Figure 2.3).
Figure 2.3 Example of a stress–strain diagram [CAM 53]
Some examples of static and dynamic ultimate strengths are given in Table 2.2.
Table 2.2. Properties of static and dynamic traction [KAR 50]
2.4. Elements of a mechanical system
In this section, we will consider lumped parameter systems, in which each particular component can be identified according to its properties and can be distinguished from other elements (in distinction from distributed systems).
Three fundamental passive elements can be defined, each playing a role in linear mechanical systems which correspond to the coefficients of the expressions of the three types of forces which are opposed to the movement (these parameters can be identified for systems with rotary or translatory movements). These passive elements are frequently used in the modeling of structures to represent a physical system in simple terms [LEV 76].
2.4.1. Mass
A mass is a rigid body whose acceleration is, according to Newton’s law, proportional to the resultant F of all the forces acting on this body [CRE 65]:
[2.9]
This is a characteristic of the body.
In the case of rotational movement, the displacement has the dimension of an angle, and acceleration can be expressed in rad/s2. The constant of proportionality is then called the moment of inertia of the body, not mass, although it obeys the same definition. The moment of inertia has the dimension M L. The inertia moment Γ is such that:
[2.10]
where Iθ is the moment of inertia and θ the angular displacement. If is the angular velocity we have:
[2.11]...
