Diferential Equations Forced vibrations ITI 26/03/2020 ITI - - PowerPoint PPT Presentation

Forced vibrations with damping Forced vibrations without damping

Diferential Equations

Forced vibrations ITI 26/03/2020

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping

1

Forced vibrations with damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

2

Forced vibrations without damping

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

Consider a sping-mass system driven by external force. Damping is present. The equation of motion is: mx′′(t) + γx′(t) + kx(t) = F(t)

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

Consider a sping-mass system driven by external force. Damping is present. The equation of motion is: mx′′(t) + γx′(t) + kx(t) = F(t) Assume F(t) is periodic: mx′′(t) + γx′(t) + kx(t) = F0 cos(ωt)

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

The general solution is: x(t) = c1x1(t) + c2x2(t)

• transient solution

+ A cos(ωt) + B sin(ωt)

• steady-state solution

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

The general solution is: x(t) = c1x1(t) + c2x2(t)

• transient solution

+ A cos(ωt) + B sin(ωt)

• steady-state solution

The transient solution is the solution of the homogeneous equation and when damping is present, as we currently assume, it goes to zero as t → ∞.

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

The general solution is: x(t) = c1x1(t) + c2x2(t)

• transient solution

+ A cos(ωt) + B sin(ωt)

• steady-state solution

The transient solution is the solution of the homogeneous equation and when damping is present, as we currently assume, it goes to zero as t → ∞. The steady state solution (also called forced responce) is: X(t) = A cos(ωt) + B sin(ωt) = R cos(ωt − δ) Through time the forced response has has constant amplitude R.

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

The amplitude, R, of the forced response depends in an interesting way on the interplay between the natural frequency ω0 =

• k/m

and the frequency of the external force ω. Steady-state amplitude R = F0

• m2(ω2

0 − ω2)2 + γ2ω2

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

The amplitude, R, of the forced response depends in an interesting way on the interplay between the natural frequency ω0 =

• k/m

and the frequency of the external force ω. Steady-state amplitude R = F0

• m2(ω2

0 − ω2)2 + γ2ω2

With the notation Γ = γ2/mk remember that

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

The amplitude, R, of the forced response depends in an interesting way on the interplay between the natural frequency ω0 =

• k/m

and the frequency of the external force ω. Steady-state amplitude R = F0

• m2(ω2

0 − ω2)2 + γ2ω2

With the notation Γ = γ2/mk remember that When Γ > 4 the system is overdamped

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

The amplitude, R, of the forced response depends in an interesting way on the interplay between the natural frequency ω0 =

• k/m

and the frequency of the external force ω. Steady-state amplitude R = F0

• m2(ω2

0 − ω2)2 + γ2ω2

With the notation Γ = γ2/mk remember that When Γ > 4 the system is overdamped When Γ = 4 the system is critically damped

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

The amplitude, R, of the forced response depends in an interesting way on the interplay between the natural frequency ω0 =

• k/m

and the frequency of the external force ω. Steady-state amplitude R = F0

• m2(ω2

0 − ω2)2 + γ2ω2

With the notation Γ = γ2/mk remember that When Γ > 4 the system is overdamped When Γ = 4 the system is critically damped When Γ < 4 the system is underdamped

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

By studying dR/dω we find that:

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

By studying dR/dω we find that: If Γ ≥ 2 the external frequence at which the steady-state amplitude has a maximum is ωmax = 0

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

By studying dR/dω we find that: If Γ ≥ 2 the external frequence at which the steady-state amplitude has a maximum is ωmax = 0 If Γ < 2 the maximum amplitude results at the following external frequency External frequency for maximum response ωmax = ω0

• 1 − Γ

2 The maximum steady-state amplitude is Rmax = F0 γω0

• 1 − Γ

4

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

For small γ ∼ 0 the forced response amplitude is quite large when ω ∼ ω0 even for small external forces; this is resonance.

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance

For small γ ∼ 0 the forced response amplitude is quite large when ω ∼ ω0 even for small external forces; this is resonance. Resonance could be catastrophically bad: Tacoma bridge ...

• r very useful: guitars,

antenas, microwave ovens ...

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping

Consider now the case without damping, i.e. γ = 0. The equation of motion is mx′′ + kx = F0 cos(ωt)

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping

Consider now the case without damping, i.e. γ = 0. The equation of motion is mx′′ + kx = F0 cos(ωt) The general solution is: x(t) = c1 cos(ω0t) + c2 sin(ω0t) + F0 m(ω2

0 − ω2) cos(ωt)

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping

Consider now the case without damping, i.e. γ = 0. The equation of motion is mx′′ + kx = F0 cos(ωt) The general solution is: x(t) = c1 cos(ω0t) + c2 sin(ω0t) + F0 m(ω2

0 − ω2) cos(ωt)

Say x(0) = 0 = x′(0). Then: x(t) = F0 m(ω2

0 − ω2)(cos(ωt) − cos(ω0t))

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping

Using trig identities, the solution becomes x(t) =

• F0

m(ω2

0 − ω2) sin (ω0 − ω)t

2

• sin (ω0 + ω)t

2

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping

Using trig identities, the solution becomes x(t) =

• F0

m(ω2

0 − ω2) sin (ω0 − ω)t

2

• sin (ω0 + ω)t

2 If ω ∼ ω0 then ω0 − ω ∼ 0 and ω0 + ω is much larger. The periodic variation of the amplitude is called a beat.

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping

Using trig identities, the solution becomes x(t) =

• F0

m(ω2

0 − ω2) sin (ω0 − ω)t

2

• sin (ω0 + ω)t

2 If ω ∼ ω0 then ω0 − ω ∼ 0 and ω0 + ω is much larger. The periodic variation of the amplitude is called a beat.

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping

Finally let us consider the case without damping, γ = 0 and with ω = ω0 =

• k/m

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping

Finally let us consider the case without damping, γ = 0 and with ω = ω0 =

• k/m

The equations of motion and simple initial conditions are: mx′′ + kx = F0 cos(ω0t), x(0) = 0 = x′(0)

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping

Finally let us consider the case without damping, γ = 0 and with ω = ω0 =

• k/m

The equations of motion and simple initial conditions are: mx′′ + kx = F0 cos(ω0t), x(0) = 0 = x′(0) The solution is: x(t) = F0 2ω0 t sin(ω0t)

ITI Forced Vibrations

Forced vibrations with damping Forced vibrations without damping

Finally let us consider the case without damping, γ = 0 and with ω = ω0 =

• k/m

The equations of motion and simple initial conditions are: mx′′ + kx = F0 cos(ω0t), x(0) = 0 = x′(0) The solution is: x(t) = F0 2ω0 t sin(ω0t)

ITI Forced Vibrations