Section 2.9. Forced Vibration questions
Objective:
1. Setting up and solve a forced vibration question
2. Classify different type of a forced vibration questions.
In this section, we finish up this chapter and solve more general spring vibration questions. Recall the spring vibration question can be solved using the 2nd order non-homogeneous differential equation, [latex]ms''(t)+rs'(t)+ks(t)=F(t)[/latex] where [latex]m[/latex] is the mass of the object hanging from the spring, [latex]r[/latex] is the damping force constant, [latex]k[/latex] is the spring constant according to Hook’s law and [latex]F(t)[/latex] is the external force on the the spring. From previous section, we know the general solution of [latex]ms''(t)+rs'(t)+ks(t)=F(t)[/latex] is [latex]s(t)=s_{c}(t)+s_{p}(t)[/latex]. Due to [latex]m,r,[/latex] and [latex]k[/latex] are all positive, the complementary solution [latex]s_{c}(t)[/latex] will approach 0 when [latex]r\gt0[/latex] and small which is the normal situation. [latex]s_{p}(t)[/latex] will present the steady state solution or the force respond. In general, [latex]F(t)=C\text{cos}(wt)[/latex] or [latex]F(t)=C\text{sin}(wt)[/latex], hence [latex]s_{p}(t)=A\text{cos}(wt)+B\text{sin}(wt)[/latex].
Example 1: A spring is stretched 20 cm by a force of 5 N. A mass of 5 kg is hung from the spring and the air resistance exerts a force of 0.1 N when the velocity of the mass is 10 m/s. There is an external force, [latex]20\text{sin}(t/2)[/latex] N, acts on the mass. If the mass is pulled down 10 cm below its equilibrium position given an initial downward velocity 10cm/s. Set up an IVP to determinate the position [latex]s(t)[/latex] of the mass at any time [latex]t[/latex]. Find steady state solution.
Exercise 1: A spring is stretched 10 cm by a force of 6 N. A mass of 4 kg is hung from the spring and the air resistance exerts a force of 0.2 N when the velocity of the mass is 20 m/s. There is an external force, [latex]20\text{cos}(t/4)[/latex] N, acts on the mass. If the mass is pulled down 5 cm below its equilibrium position given an initial downward velocity 20 cm/s. Set up an IVP to determinate the position [latex]s(t)[/latex] of the mass at any time [latex]t[/latex]. Find steady state solution.
When the damping force is zero, i.e., [latex]r=0[/latex], we have a spring system[latex]ms''(t)+ks(t)=C\text{cos}(wt)[/latex]. There is a special situation, the natural frequency of the spring is the same as the frequency of the external force, i.e., [latex]w_{0}=w[/latex] where [latex]w_{0}=\sqrt{\frac{k}{m}}[/latex]. We call this special kind of the situation resonance.
Example 2: A spring is stretched 20 cm by a force of 5 N. A mass of 5 kg is hung from the spring. There is an external force, [latex]20\text{sin}(\sqrt{5}t)[/latex] N, acts on the mass. If the mass is pulled down 10 cm below its equilibrium position given an initial downward velocity 10cm/s, determinate the position [latex]s(t)[/latex] of the mass at any time [latex]t[/latex].
Exercise 2: A spring is stretched 10 cm by a force of 6 N. A mass of 4 kg is hung from the spring. There is an external force, [latex]20\text{cos}(\sqrt{15}t)[/latex] N, acts on the mass. If the mass is pulled down 5 cm below its equilibrium position given an initial downward velocity 20 cm/s, determinate the position [latex]s(t)[/latex] of the mass at any time [latex]t[/latex].
Example 3: A mass spring system is described by the equation [latex]5s''(t)+rs'(t)+80s(t)=F(t)[/latex].
(a) When [latex]r=0[/latex], and [latex]F(t)=0[/latex], what is the system’s natural period?
(b) When [latex]r=40[/latex], and [latex]F(t)=0[/latex], find the displacement [latex]s(t)[/latex] that satisfies [latex]s(0)=6[/latex] and [latex]s'(0)=-4[/latex].
(c) When [latex]r=40[/latex], and [latex]F(t)=0[/latex], is the system underdamped, overdamped, or critical damped?
(d) True or False, regardless of the initial conditions, no solution of the system described in (b) can cross the equilibrium position more than once.
(e) When [latex]r=0[/latex], and [latex]F(t)=5\text{cos}(2t)[/latex], is the system under going resonance? How about [latex]F(t)=6\text{sin}(4t)[/latex]?
Exercise 3: A mass spring system is described by the equation [latex]2s''(t)+rs'(t)+50s(t)=F(t)[/latex].
(a) When [latex]r=0[/latex], and [latex]F(t)=0[/latex], what is the system’s natural period?
(b) When [latex]r=10[/latex], and [latex]F(t)=0[/latex], find the displacement [latex]s(t)[/latex] that satisfies [latex]s(0)=0[/latex] and [latex]s'(0)=-1[/latex].
(c) When [latex]r=10[/latex], and [latex]F(t)=0[/latex], is the system underdamped, overdamped, or critical damped?
(d) True or False, regardless of the initial conditions, no solution of the system described in (b) can cross the equilibrium position more than once.
(e) When [latex]r=0[/latex], and [latex]F(t)=2\text{cos}(5t)[/latex], is the system under going resonance? How about [latex]F(t)=6\text{sin}(5t)[/latex]?