Dynamics of Structures

Chapter 3. Response of SDOF to Harmonic Loading

Chapter 2
3.1 Response of UDSDOF to Harmonic Excitation3.2 Response of DSDOF to Harmonic Excitation3.3 Maximum Deformation and Phase Shift3.4 Rotating Unbalance3.5 Examples

3.3 Maximum Deformation and Phase Shift

Since the transient part of the vibration will eventually vanish, let's now focus on the steady state solution. It is important to understand the conditions that can cause the maximum deformation in the steady state solution in order to avoid them.

Recal the solution as,

The solution can be found as,

u(t)=e−ζωt(Acos⁡(ωDt)+Bsin⁡(ωDt))⏟Transient Solution +u(t) = \underbrace{e^{-\zeta \omega t} (A \cos(\omega_D t) + B \sin(\omega_D t)) }_{\text{Transient Solution}} \space +
C2sin⁡(ωˉt)+C3cos⁡(ωˉt)⏟Steady State Solution \underbrace{C_2 \sin(\bar{\omega} t) + C_3 \cos(\bar{\omega} t)}_{\text{Steady State Solution}}

Eq. 3.2.8

The steady state part is,

up(t)=C2sin⁡(ωˉt)+C3cos⁡(ωˉt)u_p(t) = C_2 \sin(\bar{\omega} t) + C_3 \cos(\bar{\omega} t)

Eq. 3.2.3

Which can be written in the form,

up(t)=Usin⁡(ωˉt−ϕ)u_p(t) = U \sin(\bar{\omega} t - \phi)

Eq. 3.3.1

Where,

U=Fo/k(1−r2)2+(2ζr)2U = \frac{F_o/k}{\sqrt{(1-r^2)^2 + (2 \zeta r)^2}}

Eq. 3.3.2

Or,

U=ust(1−r2)2+(2ζr)2U = \frac{u_{st}}{\sqrt{(1-r^2)^2 + (2 \zeta r)^2}}

Eq. 3.3.4

Where ust=Fo/ku_{st} = F_o \text{/} k is the displacement of the amplitude of the excitation force if it is applied statically, so it is normally called the static displacement. And the phase angle ϕ \space \phi (also calledd the phase shift) can be defined as,

ϕ=tan⁡−12ζr(1−r2)2\phi = \tan^{-1} {\frac{2 \zeta r}{(1-r^2)^2 }}

Eq. 3.3.5

Where rr is the damping ratio,

r=ωˉωr = \frac{\bar{\omega}}{\omega}

Eq. 3.1.8

The ratio of the dynamic amplitude UU to the static response ust u_{st} give rise to a very important quantity defined as the deformation response factor Rd R_d,

Rd=Uust=1(1−r2)2+(2ζr)2R_d = \frac{U}{u_{st}} = \frac{1}{\sqrt{(1-r^2)^2 + (2 \zeta r)^2}}

Eq. 3.3.6

Observations

Response Deformation Factor Rd R_d is like a multiplier of the dynamic effect from the static effect.

When the frequency ratio r=1r = 1, the forcing frequency is equal to the system frequency and system is said to be at resonance. The amplitude of the steady state solution tends to infinity, however, in real systems, failure will occur long before the system reaches very high amplitudes.

ζ\zeta

rr

ζ\zeta

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