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

3.1 Response of Undamped SDOF System to Harmonic Loading

3.1.1 Dynamic Excitation

In this section we will be exploring the response of a SDOF system under harmonic excitation. We will begin with systems subjected to harmonic excitation without daming and then we will extend the solution to include damping effects.

3.1.2 Developing and Solving the Equation of Motion

Figure 3.1.1 Mass-spring System in Forced Vibration

Figure 3.1.1 shows an undamped mass-spring system subjected to a force excitation. Let's assume that the undamped system is subjected to a harmonic excitation in the form Fo sin(ωˉt)F_o \space sin({\bar{\omega}}t).

The equation of motion can be written as,

mu¨+ku=Fosin⁡(ωˉt)m \ddot{u} + ku = F_o \sin(\bar{\omega} t)

Eq. 3.1.1

The solution can be found as,

u(t)=uocos⁡(ωt)+(voω−(Fo/k)r1−r2)sin⁡(ωt)⏟Transient Solution +u(t) = \underbrace{u_o \cos(\omega t) + \left(\frac{v_o}{\omega} - \frac{({F_o}/{k}) r}{1 - r^2}\right) \sin(\omega t)}_{\text{Transient Solution}} \space +
Fo/k1−r2sin⁡(ωˉt)⏟Steady State Solution \underbrace{\frac{{F_o}/{k}}{1 - r^2} \sin(\bar{\omega} t)}_{\text{Steady State Solution}}

Eq. 3.1.12

Where rr is the damping ratio,

r=ωˉωr = \frac{\bar{\omega}}{\omega}

Eq. 3.1.8

In absence of damping, the transient part of the solution will not vanish and will be added gradually to the steady state leading to infinite amplitude. In real systems, damping always exist and will make the transient part of the solution vansih leaving only the steady state part that goes as long as the applied harmonic excitation exist.

The equation of motion can be written as,

mu¨+ku=Fosin⁡(ωˉt)m \ddot{u} + ku = F_o \sin(\bar{\omega} t)

Eq. 3.1.1

This is a nonhomogenous differential equation and its solution is two parts, complementary solution ucu_c and and a particular solution upu_p. The complementary solution is the solution of the homogeneous equation which we solved before as,

uc(t)=Acos⁡(ωt)+Bsin⁡(ωt)u_c(t) = A \cos(\omega t) + B \sin(\omega t)

Eq. 3.1.2

he particular solution will depend on the excitation force. For a sin excitation force, the solution can take the form,

up(t)=C1sin⁡ωˉtu_p(t) = C_1 \sin{\bar{\omega} t}

Eq. 3.1.3

Differentiating with respect to time once,

u˙p(t)=C1ωˉcos⁡(ωˉt)\dot{u}_p(t) = C_1 \bar{\omega} \cos(\bar{\omega} t)

Eq. 3.1.4

Differentiating with respect to time once more,

u¨p(t)=−C1ωˉ2sin⁡(ωˉt)\ddot{u}_p(t) = -C_1 \bar{\omega}^2 \sin(\bar{\omega} t)

Eq. 3.1.5

Substituting the particular solution in the differential equation,

−mωˉ2C1+kC1=Fo-m \bar{\omega}^2 C_1 + k C_1 = F_o

Eq. 3.1.6

The constant ucu_c and and a particular solution C1C_1 can be obtained as,

C1=Fok−mωˉ2=Fo/k1−r2C_1 = \frac{F_o}{k - m \bar{\omega}^2} = \frac{{F_o}/{k}}{1 - r^2}

Eq. 3.1.7

Where rr is the damping ratio,

r=ωˉωr = \frac{\bar{\omega}}{\omega}

Eq. 3.1.8

The complete solution can be give as,

u(t)=Acos⁡(ωt)+Bsin⁡(ωt) +u(t) = A \cos(\omega t) + B \sin(\omega t) \space +
Fok(11−r2)sin⁡(ωˉt) \frac{F_o}{k} \left(\frac{1}{1 - r^2}\right) \sin(\bar{\omega} t)

Eq. 3.1.9

The two constants A and BA \space \text{and} \space B can be obtained from the initial conditions,

Using t=0,u=uot = 0, u=u_o yields,

A=uoA = u_o

Eq. 3.1.10

Using t=0,v=vot = 0, v=v_o yields,

B=voω−(Fo/k)r1−r2B = \frac{v_o}{\omega} - \frac{({F_o}/{k}) r}{1 - r^2}

Eq. 3.1.11

The solution then becomes,

u(t)=uocos⁡(ωt)+(voω−(Fo/k)r1−r2)sin⁡(ωt)⏟Transient Solution +u(t) = \underbrace{u_o \cos(\omega t) + \left(\frac{v_o}{\omega} - \frac{({F_o}/{k}) r}{1 - r^2}\right) \sin(\omega t)}_{\text{Transient Solution}} \space +
Fo/k1−r2sin⁡(ωˉt)⏟Steady State Solution \underbrace{\frac{{F_o}/{k}}{1 - r^2} \sin(\bar{\omega} t)}_{\text{Steady State Solution}}

Eq. 3.1.12

In absence of damping, the transient part of the solution will not vanish and will be added gradually to the steady state leading to infinite amplitude. In real systems, damping always exist and will make the transient part of the solution vansih leaving only the steady state part that goes as long as the applied harmonic excitation exist.

Observations

In real systems, damping will always exist and the transient part will always vanish after a certain period of time.

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.

uou_o

vov_o

ω\omega

ωˉ\bar{\omega}

FoF_o

kk

tt