To ensure that the amplitude of a damped driven harmonic oscillator changes with time linearly, you need to apply a driving force that varies with time as a linear function. Specifically, the driving force should have a time-dependent component that increases or decreases linearly with time.
Mathematically, if we assume the equation of motion for the damped driven harmonic oscillator is given by:
m * d^2x/dt^2 + c * dx/dt + k * x = F(t),
where m is the mass of the oscillator, c is the damping coefficient, k is the spring constant, x is the displacement of the oscillator, and F(t) is the driving force.
To achieve a linearly changing amplitude, the driving force F(t) should be of the form:
F(t) = a * t + F0,
where a is a constant representing the rate of change of the driving force, t is time, and F0 is an offset constant.
By choosing a linearly increasing or decreasing driving force, the amplitude of the oscillator will also change linearly with time. The constant a determines the rate of change of the amplitude. If a is positive, the amplitude will increase linearly; if a is negative, the amplitude will decrease linearly.
Please note that the above analysis assumes a simple linear driving force. In practice, the equation of motion for a damped driven harmonic oscillator can involve more complex driving forces, such as sinusoidal or other periodic functions. The linear driving force mentioned here is a simplification to achieve a linearly changing amplitude.