Introduction to acoustics

Part 1 : I - Free Vibrations 1 DDL The vast majority of the elementary mechanical system are based on a simple but crucial notion : Degree of freedom (DOF) In this part of the course we’ll be talking about the first DOF, which mean that the only mass of the system is allowed to move in only one direction. In the case a system can move in various direction it’s called a system with a various DOF, it’s as simple as that. In this part 2 types of system will be encounter : in translation and in torsion Translation That kind of movement may be represented by a mass [kg] / spring [N/m] system which equation is : F= -k.x Thanks to the balance equation the differential equation may be easily found : m¨x + kx And it’s solution x(t)=Asin(ωt+φ) With A -> Amplitude ω-> Phase φ->Pulse The natural pulse is the value that satisfy this very next equation : ω0= √k/m =2πf0 [rad/s] Note : Hanging system -> resting mass carry out a traction force mgon the spring that may be traduced by an initial stretch x0 In that case a new movement equation is found : m¨x + mg + kx + kx 0= 0 Torsion A rigid body oscillate around an axis (torsion vibration) Dynamic balance between dynamic moment and torsion moment allowed us to write the angular movement equation : J0¨Θ + k Θ = 0 With J0 Inertia moment A small but significant change occur on the natural pulse equation in this system : ω=√k/J0 [rad/s] Equivalent stiffness When it’ll come the time of resolving problem one thing to know is essential : How to add up springs. Parallel Springs F=k1x + k2x = keqx Generally : keq=k1+k2+...+kn Series Springs 1/keq=1/k1+1/k2 Generally : 1keq= 1/k1+1/k2+...+1/kn Linearization Simple gravity pendulum Surely the most simple example in mechanic that is widely used and known by student. It consist in an idealization of a real pendulum using such assumptions as : (https://en.wikipedia.org/wiki/Pendulum_%28mathematics%29) The rod or cord on which the bob swings is massless, inextensible and always remains taut; The bob is a point mass; Motion occurs only in two dimensions, i.e. the bob does not trace an ellipse but an arc. The motion does not lose energy to friction or air resistance. The gravitational field is uniform. The support does not move What it’s the most important to remember in our case ? Linearization consider sinθ≈θ for small movement which mean that the differential equation link to this system is : J0+mglθ = 0 And it’s natural pulse become : ω0=√g/l Compound pendulum In opposition of the precedent example the mass is not considered anymore as a point and the motion is perfect no more. Which mean some changes in the differential equation and natural pulse : J0¨θ+Mgdθ=0 (Linearized equation) ω0 =√Mgd/J0

Part 2 : II - Forced Response