Response of DSDOF Systems to Harmonic Loading

Chapter 3. Response of SDOF to Harmonic Loading

Chapter 2

3.1 Response of UDSDOF to Harmonic Excitation 3.2 Response of DSDOF to Harmonic Excitation 3.3 Maximum Deformation and Phase Shift 3.4 Rotating Unbalance 3.5 Examples

3.2 Response of Damped SDOF System to Harmonic Loading

3.2.1 Developing and Solving the Equation of Motion

Figure 3.2.1 Mass-spring of Damped System in Forced Vibration

Figure 3.2.1 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 $F_o \space sin({\bar{\omega}}t)$ .

The equation of motion can be written as,

m \ddot{u} + c \dot{u} + ku = F_o \sin(\bar{\omega} t)

Eq. 3.2.1

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

u_c(t) =e^{-\zeta \omega t} (A \cos(\omega_D t) + B \sin(\omega_D t))

Eq. 3.2.2

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

u_p(t) = (C_2 \sin(\bar{\omega} t) + C_3 \cos(\bar{\omega} t))

Eq. 3.2.3

Differentiating for the first and second derivative, the constants can be obtained as,

C_2 = (F_o/k) \frac{(1-r^2)}{(1-r^2)^2 + (2 \zeta r)^2}

Eq. 3.2.4

C_3 = (F_o/k) \frac{-2 \zeta r}{(1-r^2)^2 + (2 \zeta r)^2}

Eq. 3.2.5

The complete solution can be give as,

u(t) = e^{-\zeta \omega t} (A \cos(\omega_D t) + B \sin(\omega_D t)) +

(C_2 \sin(\bar{\omega} t) + C_3 \cos(\bar{\omega} t))

Eq. 3.2.6

The two constants $A$ and $B$ can be obtained from the initial conditions, at $t=0, u = u_o$ and $v = v_o$ .

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

A = u_o -C_3

Eq. 3.2.7

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

B = \frac{v_o + \zeta \omega A - C_2 \bar{\omega} }{\omega_D}

Eq. 3.2.7

The solution can be found as,

u(t) = \underbrace{e^{-\zeta \omega t} (A \cos(\omega_D t) + B \sin(\omega_D t)) }_{\text{Transient Solution}} \space +

\underbrace{C_2 \sin(\bar{\omega} t) + C_3 \cos(\bar{\omega} t)}_{\text{Steady State Solution}}

Eq. 3.2.8

The transient part of the solution will vansih leaving only the steady state part that goes as long as the applied harmonic excitation exist.

Observations

Damping will make the transient part to vanish after a certain period of time leaving only the steady state park of the solution.

When the frequency ratio $r = 1$ , 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.

$u_o$

$v_o$

$\omega$

$\bar{\omega}$

$F_o$

$k$

$\zeta$

$t$

Dynamics of Structures

Structural Engineering Simplified

Chapter 3. Response of SDOF to Harmonic Loading

Chapter 2

3.2 Response of Damped SDOF System to Harmonic Loading

3.2.1 Developing and Solving the Equation of Motion