Indicate whether each of the following statements is true or false: a. The equat
ID: 1718191 • Letter: I
Question
Indicate whether each of the following statements is true or false: a. The equation of motion of a single-degree-of-freedom system will be the same whether the mass moves in a horizontal plane or an inclined plane. b. The principle of conservation of energy can be used to derive the equation of motion of both damped and undamped systems. 2. The equation of motion of a spring-mass-damper system is expressed as 4x+16x+16x=0 using SI unit system, find the following: 1). mass m, spring constant k, and damping constant c 2). circular natural frequency omegan, damping coefficient zeta, and damped natural frequency omegad 3). the characteristic equation of the system 4). the response of the system x(t) 5). with initial condition chi0 = 1, and chi0 = chi0 ,find at time t = 1second: (a) displacement chi(1), and (b) velocity chi(l)Explanation / Answer
1 a) True , gravitational forces acting on the mass have no effect on the equation of motion.
b) Flase, Conservation of energy can be used only to derive motions of undamped systems.
2 a) compare equation with md^x/dt^2 + cdx/dt + k x = 0
we get m = 4 , c = 16 , k = 16
2 b) Wn = sqrt(k/m) = 2
damping ratio (zeta) = c / 2*sqrt(k*m) = 16 / 2 * sqrt(4 * 16) = 1
Wd = Wn * sqrt(1 - zeta^2 ) = 0
2 c) characteristic equation d^x/dt^2 + 4dx/dt + 4x = 0
d) x(t) = (c1 + c2*t) * exp(-Wn*t)
e) x(t) = (1 + 2*t)*exp(-2*t)
x(1) = 3*exp(-2)
dx(1)/dt = -4*exp(-2)