Vibrations in linear 1 degree of freedom systems

Vibrations in linear 1 -degree of freedom systems; I. undamped systems (last updated 2011 -08 -26) Kjell Simonsson 1

Aim The aim of this presentation is to give a short review of basic vibration analysis in undamped linear 1 degree-of-freedom (1 -dof) systems. For a more comprehensive treatment of the subject, see any book in basic Solid Mechanics or vibration analysis. Kjell Simonsson 2

Linear 1 -dof systems A 1 -dof system is a simplification of reality, where it has been assumed that the mass of the body/structure • can be associated with one specific small part of the body • is restricted to one "major vibration mode" A typical example of a situation for which a 1 -dof assumption is applicable is a slender structure on which a single heavy detail is attached, where the latter mainly moves in one (translational or angular) direction, see below where the movement is depicted in blue. The term linear 1 dof-systems, implies that all the governing relations are linear, which e. g. is the case of a linear elastic structure (material linearity), subjected to so small deformations that the ordinary stress and strain measures may be used (geometric linearity). Kjell Simonsson 3 lateral motion horizontal motion rotational motion

A simple example Let us study some basic phenomena of linear 1 -dof vibration analysis (of un-damped) systems, by considering one of the simple examples shown above! By making a free body diagram of the (point) mass, we get where the coordinate x describes the position of the mass (equal to the deflection of the rod), and where the force S represents the interaction between the rod and the point mass. Elementary solid mechanics now gives Kjell Simonsson 4

A simple example; cont. By introducing the lateral/bending stiffness k, we get or, by dividing with m and introducing the complete expression for F Thus, the lateral motion of the mass as a function of time is governed by a linear 2'nd order ordinary differential equation with constant coefficients! The solution is given by the sum of the homogeneous solution and the particular solution, , where the former corresponds to the self vibration of the mass caused by the initial conditions (placement and velocity), while the latter is caused by the applied force. Kjell Simonsson 5

A simple example; cont. Homogeneous solution For the homogeneous solution we have As can be seen, for free vibrations with F=0, the point mass can only vibrate with the frequency , which we refer to as the natural frequency or eigenfrequency of the structure. By denoting it , we get Kjell Simonsson 6

A simple example; cont. Homogeneous solution; cont. Alternatively we may express this as A & B or X & ϕ are found by using the initial conditions Since we in all physical contexts have some kind of damping, i. e. processes that transform the mechanical energy into e. g heat, it follows that the self vibration caused by some initial conditions sooner or later will vanish. Thus, at such a stationary state ("fortvarighet" in Swedish), only the particular solution will remain. It can finally be noted that the circular eigenfrequency fe and the eigenperiod Te are defined as Kjell Simonsson 7

Resonance Homogeneous solution; cont. We may also analyze the eigenfrequency of a structure/body by the Finite Element method (FEM), see below where an animation of the vibration mode (eigenmode) can be found for a similar type of structure. Kjell Simonsson 8

A simple example; cont. Particular solution For the particular solution we let As can be seen, forced vibrations the displacement amplitude will depend on how close to the eigenfrequency the applied frequence is. Theoretically, we will get infinite amplitudes for the case. We refer to this as resonance! Kjell Simonsson 9

A simple example; cont. Particular solution; cont. It may be noted that the particular solution can be recast in another form, as illustrated below where the factor is the so called dynamic impact factor Kjell Simonsson 10

Resonance As we saw above, the amplitude of the particular solution goes to infinity when the applied frequency approaches the eigenfrequency of the system. Since the obtained particular solution is not valid for this case (we are not allowed to divide by zero), we instead have to proceed as follows Kjell Simonsson 11

Resonance On the previous slide we found that the amplitude of the forced vibration at resonance grows linearly with time i. e. we get the type of behavior illustrated to the right Even though (as will be seen later on) damping will make theoretical vibration amplitude finite, it may still be very large near the "resonance" frequency. The same is true for multi-dof systems and continuous systems. As a consequence, all engineering designs must be designed such that they are not operated in a resonance region, since that inevitably will cause premature failure unless additional damping devices are used. Kjell Simonsson 12