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 Forced vibrations with damping 1 Driven systems General, transient and steady-state solution The amplitude of the forced response Resonance Forced vibrations without damping 2 ITI Forced Vibrations
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping 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 ) + γ x ′ ( t ) + kx ( t ) = F ( t ) ITI Forced Vibrations
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping 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 ) + γ x ′ ( t ) + kx ( t ) = F ( t ) Assume F ( t ) is periodic: mx ′′ ( t ) + γ x ′ ( t ) + kx ( t ) = F 0 cos( ω t ) ITI Forced Vibrations
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping The amplitude of the forced response Resonance The general solution is: x ( t ) = c 1 x 1 ( t ) + c 2 x 2 ( t ) + A cos( ω t ) + B sin( ω t ) � �� � � �� � transient solution steady-state solution ITI Forced Vibrations
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping The amplitude of the forced response Resonance The general solution is: x ( t ) = c 1 x 1 ( t ) + c 2 x 2 ( t ) + A cos( ω t ) + B sin( ω t ) � �� � � �� � transient solution 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
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping The amplitude of the forced response Resonance The general solution is: x ( t ) = c 1 x 1 ( t ) + c 2 x 2 ( t ) + A cos( ω t ) + B sin( ω t ) � �� � � �� � transient solution 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( ω t ) + B sin( ω t ) = R cos( ω t − δ ) Through time the forced response has has constant amplitude R . ITI Forced Vibrations
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping 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 ω 0 = k / m and the frequency of the external force ω . Steady-state amplitude F 0 R = � 0 − ω 2 ) 2 + γ 2 ω 2 m 2 ( ω 2 ITI Forced Vibrations
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping 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 ω 0 = k / m and the frequency of the external force ω . Steady-state amplitude F 0 R = � 0 − ω 2 ) 2 + γ 2 ω 2 m 2 ( ω 2 With the notation Γ = γ 2 / mk remember that ITI Forced Vibrations
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping 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 ω 0 = k / m and the frequency of the external force ω . Steady-state amplitude F 0 R = � 0 − ω 2 ) 2 + γ 2 ω 2 m 2 ( ω 2 With the notation Γ = γ 2 / mk remember that When Γ > 4 the system is overdamped ITI Forced Vibrations
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping 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 ω 0 = k / m and the frequency of the external force ω . Steady-state amplitude F 0 R = � 0 − ω 2 ) 2 + γ 2 ω 2 m 2 ( ω 2 With the notation Γ = γ 2 / mk remember that When Γ > 4 the system is overdamped When Γ = 4 the system is critically damped ITI Forced Vibrations
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping 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 ω 0 = k / m and the frequency of the external force ω . Steady-state amplitude F 0 R = � 0 − ω 2 ) 2 + γ 2 ω 2 m 2 ( ω 2 With the notation Γ = γ 2 / 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
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping The amplitude of the forced response Resonance By studying dR / d ω we find that: ITI Forced Vibrations
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping The amplitude of the forced response Resonance By studying dR / d ω we find that: If Γ ≥ 2 the external frequence at which the steady-state amplitude has a maximum is ω max = 0 ITI Forced Vibrations
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping The amplitude of the forced response Resonance By studying dR / d ω we find that: If Γ ≥ 2 the external frequence at which the steady-state amplitude has a maximum is ω max = 0 If Γ < 2 the maximum amplitude results at the following external frequency External frequency for maximum response � 1 − Γ ω max = ω 0 2 The maximum steady-state amplitude is F 0 R max = � 1 − Γ γω 0 4 ITI Forced Vibrations
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping The amplitude of the forced response Resonance For small γ ∼ 0 the forced response amplitude is quite large when ω ∼ ω 0 even for small external forces; this is resonance. ITI Forced Vibrations
Driven systems Forced vibrations with damping General, transient and steady-state solution Forced vibrations without damping The amplitude of the forced response Resonance For small γ ∼ 0 the forced response amplitude is quite large when ω ∼ ω 0 even for small external forces; this is resonance. Resonance could be catastrophically bad: Tacoma bridge ... or 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 = F 0 cos( ω t ) 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 = F 0 cos( ω t ) The general solution is: F 0 x ( t ) = c 1 cos( ω 0 t ) + c 2 sin( ω 0 t ) + 0 − ω 2 ) cos( ω t ) m ( ω 2 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 = F 0 cos( ω t ) The general solution is: F 0 x ( t ) = c 1 cos( ω 0 t ) + c 2 sin( ω 0 t ) + 0 − ω 2 ) cos( ω t ) m ( ω 2 Say x (0) = 0 = x ′ (0). Then: F 0 x ( t ) = 0 − ω 2 )(cos( ω t ) − cos( ω 0 t )) m ( ω 2 ITI Forced Vibrations
Forced vibrations with damping Forced vibrations without damping Using trig identities, the solution becomes � � 0 − ω 2 ) sin ( ω 0 − ω ) t F 0 sin ( ω 0 + ω ) t x ( t ) = m ( ω 2 2 2 ITI Forced Vibrations
Forced vibrations with damping Forced vibrations without damping Using trig identities, the solution becomes � � 0 − ω 2 ) sin ( ω 0 − ω ) t F 0 sin ( ω 0 + ω ) t x ( t ) = m ( ω 2 2 2 If ω ∼ ω 0 then ω 0 − ω ∼ 0 and ω 0 + 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 � � 0 − ω 2 ) sin ( ω 0 − ω ) t F 0 sin ( ω 0 + ω ) t x ( t ) = m ( ω 2 2 2 If ω ∼ ω 0 then ω 0 − ω ∼ 0 and ω 0 + 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 = ω 0 = k / m ITI Forced Vibrations
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