
In this lecture, we will again consider the equation of motion for the spring-mass-damper model. However, now it is subject to a harmonic forcing function, $P_0\sin(\omega t)$.

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/solving-the-equation-of-motion/img1.jpg"
	alt="Spring-mass-damper model subject to a harmonic force. | EngineeringSkills.com"
	width="543"
	height="239"
	caption="Fig 1. Spring-mass-damper model subject to a harmonic force."
/>

Building on our study of free vibration, the equation of motion that describes this forced system is,

$$
m\ddot{u} + c\dot{u} + ku=P_0\sin(\omega t)
\tag{1}
$$

Our objective now is to solve this differential equation to determine an expression for the displacement as a function of time, $u(t)$. This is a non-homogeneous second-order differential equation and as such its solution has two parts:

- The complementary solution (also the solution to the equivalent homogeneous equation)

- The particular solution

We've already developed the complementary solution, this is the free vibration response. So what remains now is to find the particular solution. This proceeds in the usual way with the term on the right-hand side of the equal sign suggesting the form of a trial solution.

Since the right-hand side is a sinusoid, the trial solution will be a generalised sinusoid,

$$
u(t) = A\sin(\omega t) + B\cos(\omega t)
\tag{2}
$$

We now differentiate this trial solution so it can be substituted back into our original differential equation,

$$
\dot{u}(t) = \omega A\cos(\omega t) -\omega B\sin(\omega t)
\tag{3}
$$

$$
\ddot{u}(t) = -\omega^2 A\sin(\omega t) -\omega^2 B\cos(\omega t)
\tag{4}
$$

Substituting these expressions back into our differential equation yields,

$$
\begin{split}
m\underbrace{\left[-\omega^2 A\sin(\omega t) -\omega^2 B\cos(\omega t) \right]}_{\text{acceleration, } \ddot{u}} + c\underbrace{\left[\omega A\cos(\omega t) -\omega B\sin(\omega t) \right]}_{\text{velocity, }\dot{u}} + \\ + k\underbrace{\left[A\sin(\omega t) + B\cos(\omega t) \right]}_{\text{displacement, }u} = P_0\sin(\omega t)
\end{split}
\tag{5}
$$

Rearranging yields,

$$
(k-m\omega^2)\left[A\sin(\omega t) + B\cos(\omega t)\right] + c\omega\left[A\cos(\omega t)-B\sin(\omega t)\right] = P_0\sin(\omega t)
\tag{6}
$$

We can obtain expressions for the constants $A$ and $B$ by collecting sine and cosine terms together. Collecting sine terms yields (after sine terms cancelled),

$$
A(k-m\omega^2) - B(c\omega) = P_0
\tag{7}
$$

Collecting and cancelling out cosine terms yields,

$$
A(c\omega) + B(k-m\omega^2) =0
\tag{8}
$$

These two simultaneous equations are used to obtain the following expressions for $A$ and $B$,

$$
A = \frac{k-m\omega^2}{(k-m\omega^2)^2 + (c\omega)^2} P_0
\tag{9}
$$

$$
B = \frac{-c\omega}{(k-m\omega^2)^2 + (c\omega)^2} P_0
\tag{10}
$$

Finally, these expressions can be substituted back into our assumed solution for $u(t)$, to yield the following expression for the particular solution,

$$
u_p(t) = \frac{P_0}{(k-m\omega^2)^2 + (c\omega)^2}\left[(k-m\omega^2)\sin(\omega t) - c\omega \cos(\omega t) \right]
\tag{11}
$$

Combining the complementary and particular solutions we obtain the complete expression for the response of the SDoF system under harmonic excitation,

$$
u(t) = u_c(t) + u_p(t)
\tag{12}
$$

$$
\boxed{u(t) = e^{-\xi \omega_nt}\:\left[A\sin(\omega_d t) +B\cos(\omega_d t)\right] + \frac{P_0}{(k-m\omega^2)^2 + (c\omega)^2}\left[(k-m\omega^2)\sin(\omega t)-c\omega\cos(\omega t)\right]}
\tag{13}
$$

Notice that the complementary solution describes oscillations at the damped natural frequency of the structure $\omega_d$, while the particular solution describes oscillations at the forcing frequency. As we've said, the complementary solution describes the free vibration response. This corresponds to transient behaviour that occurs at the beginning of the oscillatory behaviour when the force is first applied. This transient response decays away exponentially leaving only the response described by the particular solution.

The particular solution is therefore often referred to as the steady-state solution. For a dynamic system experiencing harmonic forcing, the steady-state solution is typically of more interest than the transient solution.

Note also that in the equation above, the transient portion of the response still contains unknown constants of integration $A$ and $B$. These can be determined from knowledge of the system initial conditions (position and velocity). But as we've said, this portion of the response decays to zero and as such will not be of interest when considering the long-term steady-state oscillating behaviour.

### _Dimensionless_ response equation

Equation 13 above that describes the complete response is often presented in an alternative form that makes use of dimensionless ratios. If we define $\beta$ as the frequency ratio,

$$
\beta = \frac{\omega}{\omega_n}
\tag{14}
$$

where $\omega$ is the forcing frequency, we can restate the response equation as,

$$
\boxed{u(t) = e^{-\xi \omega_nt}\:\left[A\sin(\omega_d t) +B\cos(\omega_d t)\right] + \frac{P_0}{k}\left[\frac{1}{(1-\beta^2)^2+(2\xi\beta)^2}\right]\left[(1-\beta^2)\sin(\omega t)-2\xi\beta\cos(\omega t)\right]}
\tag{15}
$$

In the next lecture we'll examine the steady-state response in more detail in an attempt to extract some of its defining characteristics.


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