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Mechanical System DynamicsProf. Roberto Corradi Transverse vibration of stretched strings Politecnico di MilanoM.Sc. in Mechanical Engineering 2 Model of a stretched string 1)shear force and bending moment are neglected 2)the string is made of homogeneous material and it has constant cross-section (the mass per unit length mis constant along the string) 3)the tension Tis high with respect to the string’s weight; consequently, in the static equilibrium position in figure, the string’s configuration can be approximated as rectilinear 4)small vibration amplitudes; therefore tension variations due to the string’s transverse vibration are negligible with respect to the value of Tin the static configuration (3+4 →the tension Tis assumed to be constant with xand t) 5)damping is neglected Lx T, mw(x,t) Lx T, mw(x,t) wis the transverse displacement of the string, evaluated at time tand in correspondence with the string’s section located in x. Just as an example, the pictureon the left represents a string with fixed ends. But actually, boundary conditions are not specified at this stage. 3 We are looking for the homogeneous partial differential equation that governs the string’s undamped vibration about its static equilibrium configuration. No external loads, except for concentrated (constraint or active) forces at the boundary, are supposed to act on the string. The equation we are going to derive is valid independently of the specific boundary conditions. Thus, consider a string’s element with infinitesimal length dxand plot its free-body diagram: the only forces acting on the infinitesimal string’s portion dxare the tension Tat the two extremities and the inertia force Fin. 2 2in wF mdx t =  sin sin 0in R LFT T −+ − = The string’s equation Force balance in the transverse…