9.7 Response of Beams Due to Moving Loads
389
If the beam is at rest before the load is applied, then ξ r = ˙
ξ r = 0 at t = 0 and the
constants A r and B r can be evaluated from the initial conditions.
Equation (9.84) indicates that large amplitudes will be obtained when
ω r = ω ±
r πv
L
(9.85)
In the derivation of this section, the mass of the load has been neglected.
Exercise 9
9.1 A bar with built-in ends is acted upon by a constant axial force P, applied at
quarter point, as shown in the figure. Write the equation of axial displacement
of the bar, if the bar was unstressed before P was applied.
Prob. 9.1
9.2 A drill string is made of steel and is 1400 m long. Determine the fundamental
mode of vibration by considering the string to be free at one end and clamped
at the other. Find the displacement at the free end when t = T /3, where T
is the time period due to a tensile stress σ = P/A = 21 N/mm
2 suddenly
applied to this end. E = 2.1 × 10
11 N/m
2 and ρ = 7800 kg/m
2 .
9.3 A simply supported beam of mass M is suddenly loaded by a force shown in
the figure. Determine the equation of deflection of the beam.
Prob. 9.3
9.4 A missile is excited longitudinally by the thrust F(t) of its rocket engine, at the
end x = 0 during its flight. Write a general expression for the displacement
of the missile.
389
If the beam is at rest before the load is applied, then ξ r = ˙
ξ r = 0 at t = 0 and the
constants A r and B r can be evaluated from the initial conditions.
Equation (9.84) indicates that large amplitudes will be obtained when
ω r = ω ±
r πv
L
(9.85)
In the derivation of this section, the mass of the load has been neglected.
Exercise 9
9.1 A bar with built-in ends is acted upon by a constant axial force P, applied at
quarter point, as shown in the figure. Write the equation of axial displacement
of the bar, if the bar was unstressed before P was applied.
Prob. 9.1
9.2 A drill string is made of steel and is 1400 m long. Determine the fundamental
mode of vibration by considering the string to be free at one end and clamped
at the other. Find the displacement at the free end when t = T /3, where T
is the time period due to a tensile stress σ = P/A = 21 N/mm
2 suddenly
applied to this end. E = 2.1 × 10
11 N/m
2 and ρ = 7800 kg/m
2 .
9.3 A simply supported beam of mass M is suddenly loaded by a force shown in
the figure. Determine the equation of deflection of the beam.
Prob. 9.3
9.4 A missile is excited longitudinally by the thrust F(t) of its rocket engine, at the
end x = 0 during its flight. Write a general expression for the displacement
of the missile.