A spring-mass system, resting on an inclined plane, is subjected to a harmonic force as shown in Fig. 3.38. Find the response of the system by assuming zero initial conditions.

Step 1: Derive the equation of motion.
Let $x$ be the displacement of the mass from its static equilibrium position.
The forces acting on the mass along the inclined plane are:

• The spring force: $-k(x_{st} + x)$, where $x_{st}$ is the static deflection of the spring and $x$ is the dynamic displacement.
• The component of gravity along the incline: $mg \sin \theta$.
• The applied harmonic force: $f(t) = F_0 \cos \omega t$.

At static equilibrium, the spring force balances the gravitational component:

$$kx_{st} = mg \sin \theta$$

Applying Newton's second law ($m\ddot{x} = \sum F$) for the dynamic motion, considering $x$ as the displacement from static equilibrium:

$$m\ddot{x} = F_0 \cos \omega t + mg \sin \theta - k(x_{st} + x)$$

Substitute $mg \sin \theta = kx_{st}$:

$$m\ddot{x} = F_0 \cos \omega t + kx_{st} - kx_{st} - kx$$ $$m\ddot{x} + kx = F_0 \cos \omega t$$

This is the equation of motion for an undamped, forced single-degree-of-