I would like to know the step by step method of how the general solutions in the pictures were obtained. The generalized mode mass, the displacement solution, velocity solution, internal forces

solution. For the fixed-free impacted rod/n CHAPTER 3 MATHEMATICAL MODEL 3.1. Introduction The study of stress wave propagation in elastic bodies has a long history. The elastic rod can undergo longitudinal, lateral, and torsional vibration. The longitudinal propagation of stress waves in rods has been investigated extensively over the last two centuries. Axial impact on an elastic body result in a disturbance that initially propagates away from the impact site at a specific speed. This disturbance is a pulse or wave of particle displacement (and consequent stress). Wave propagation relates to propagating a coherent pulse of stress and particle displacement through a medium at a characteristic speed. Typical manifestations of this phenomenon are the transmission of sound through air, water waves across the sea's surface, and seismic tremors through the earth, i.e., waves exist in gases, liquids, and solids. Sources of excitation may be either concentrated or distributed spatially and brief or continuous functions of time. The unifying characteristic of waves is propagating a disturbance through a medium. Properties of the medium that result in waves and determine the propagation speeds are the density p and moduli of deformability, e.g., young's modulus E, shear modulus G, bulk modulus K, etc. 3.2. Longitudinal impact of a rigid mass against an elastic rod Considering a stationary homogenous elastic rod with mass m, young modulus E, density p, variable cross-sectional area A(x), and length L. The rod struck on the right end x = L at time t = 0 by a moving rigid mass Mp of mass ratio y = mrod / Mp and initial velocity vo, see Figure 3.1. mrod, P, E Mp L "↑ 12 Tr(x) L k (b) (a) Figure 3.1: Longitudinal impact of a mass on a rod of variable cross-section; (a) impact model and (b) Shape and variables of the rod. Y. CS CamScanner 3.2.1. Assumptions for the derivation of the equation of motions We consider the following assumptions in the derivation of the wave equation of motion Plane, parallel cross sections remain plane and parallel during deformation so that the local contact region's deformation and transverse waves of the rod and vibrations of the striking mass are neglected. ● ● Frictionless contact surface The rod material is homogeneous so that E and p do not vary with x. Uniform distribution of stress across the cross section. Rod material behaves elastically (Hooke's Law). No body forces These assumptions allow the displacement to be specified as a function of one space coordinate indicating location along the rod's length. However, lateral deformation can be found in any cross-section. 3.2.2. The governing differential equation The resulting motion after impact is assumed to be one-dimensional with longitudinal displacement u(x, t), as shown in Figure 3.2. The equation of motion can be obtained from the force equilibrium equation for an element dx of the rod in x-direction [10], [15][17] as follows HIT მთ A- ax L +0 dx JA ax JA A+ dx ax Figure 3.2: An element from a variable cross-section rod. JA 1 -0A + (a +=dx) (A + x) = 0 {A+ (4 + 2dx)} dx -σA dx (A dxdx- əx at² ot = PA 13 aa dx a²u at² x (3.1) (3.2) CS CamScanner Then: Or where A(x), o (x, t), u(x, t) are the cross-sectional area, the stress across the cross section, and the longitudinal displacement, respectively. Now assume that the rod material behaves elastically, and the simple Hooke's law applies, where E is the Young's modulus and e(x, t) is the axial strain, defined by: where Co = 1 a Адх u(0, t) = 0 du(0, t) Əx du(0, t) AE əx ·(σA) = P₁ ɛ(x, t) = = = 0 By assuming that the rod material is homogeneous so that E and p do not vary with x, the equation of motion becomes: = ku(0, t) o(x, t) = E E a A dx du(x, t) əx ди (ADU) a²u at² du(x, t) əx 0x2 ²u 1 dA du + ax² Adx ax 0x2 = P = √E/p. For the rod with constant cross-section area dA/dx = 0 ²u c? 0t² ²u at² 1 0²u c²t² 1 0²u 14 (3.3) Eq. 3.8 is the wave equation of the longitudinal displacement u at the position x at time t, and . Co denotes the longitudinal wave propagation velocity of the infinitesimal elastic pulse in the rod. (when the rod at x = 0 is fixed- no displacement) (when the rod at x = 0 is free - no stress) (when the rod at x = 0 is attached with a spring) (3.4) 3.2.3. The boundary conditions The rod could have fixed, free, or elastic (spring with stiffness k attached) boundary conditions at x = 0 as follows: (3.5) - (3.6) (3.7) (3.8) (3.9) CS CamScanner At x = L, the rigid mass will remain in contact with the rod for period of contact time t, an Lis: the contact force is compressive, then the boundary condition at x = BA du(L, t) əx EA = - Mp Ju(L, t) əx After that time, the rigid mass is no longer in contact with the rod, and the rod perform free vibration without the mass at the tip. Hence: du(x,0) at = 0 a²u(L, t) at² = 0 =-vo for 3.2.4. The initial conditions According to St. Venant's contact theory, at the instant of the impact, the velocity of th struck end of the rod becomes immediately equals to the velocity of the rigid mass vo. u(x,0) = 0 for 0≤x≤L du(x, 0) at at for 0 tc 0≤x