Teoria - Dynamics of Mechanical Systems

Name: Teoria - Dynamics of Mechanical Systems
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Gli appunti spaziano sugli aspetti teorici utili al superamento dell'esame.Sono frutto di rielaborazione e reinterpretazione del publisher del materiale fornito dal professore con disegni di mia …

Esame Dynamics of mechanical systems

Facoltà Ingegneria dell'informazione

Dal corso del Prof. Bruni Stefano

Università Politecnico di Milano

A.A. 2018-2019

114 pagine

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Dynamics of lumped parameter systems

Mass, inertia, elasticity and damper divided.

Equations of motion for a mechanical system

Let us consider a mechanical system having n degree of freedom (d.o.f.) The equations of motion are a set of 2^nd order differential equations in n unknowns represented by the system's coordinate x. With a standard mathematical treatment, it is possible to end up with a set of 2n 1^st order differential equations.

Newtonian dynamics

The F=ma approach is based on Newton's second law:

F: Sum of all forces applied to the point
a: Acceleration of the point mass
F=ma
M_O=J_Ow

Newton's second law for a single rigid body in a planar motion:

a_C: Acceleration of the center of mass (C.O.M.) of the body
M_O: Moment of all forces acting on the body referred to a point O called the "centre"
J_O: Moment of inertia of the body w.r.t. the same point O
w: Angular velocity of the body

In planar motion, vectors are directed perpendicularly to the plane in which the movement takes place.

Example: 1 d.o.f. system

F_sK_X-cx-mx=0
F-kx-cx-mx
x=Fmx + cX + kX = F

D'Alembert's principle

F_in = -ma (Inertia force)
M_in = -J_O W (Moment of inertia forces)

Example:

F_d: K_x-cx-mx
F-kx-cx-mx
x= mx + cX + kX=F

We end up with the same equation of motion.

The principle of virtual work

It represents the movement of the system in the form of equilibrium equations: (SW SW_n =0)

L + L_in = for any virtual displacement

Under the assumption of non-dissipative constraints (constant forces generate zero work, frictionless constraints and rolling without sliding condition), the equation brings a set of n independent equations of motion (m d.o.f).

(This equation does not include any unknown constraint force) F_sX-(kx cx mxx)

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