Why can't resonance occur when an object is forced to vibrate (in simple harmonic motion) with a frequency other than the natural frequency?

Question

As far as I understood forced vibrations, after a point of time steady state is reached i.e. the body oscillates in accordance with the applied force irrespective of the force frequency (may be other than the natural frequency of the body).Doesn't this mean the phase difference between the applied force and the displacement/vibration in simple harmonic motion is 90 degrees at steady state?

Answer 1

NO they will be in the same phase You can get this answer by using the formula given for forced oscillation and using appropriate derivations to get your result I will give you that here-

Let an external periodic force (F=F0 sin ωt) where F0 is the amplitude of the impressed force and w is the angular frequency of the impressed force be applied on the oscillator.

The equation of motion of forced harmonic oscillations will be-

Let the solution of equation in the steady state be x2 = A sin (wt-θ). [Since in the steady state, the amplitude of forced oscillation is constant l (=A, say) and the frequency is equal to the frequency of impressed force i.e ω/ 2π.

Here θ is the phase difference between the displacement and the impressed force.

Then

$$dx_2/dt = A ω cos (ω t-θ) $$ $$ {d^2}x_2/{dt^2} = -A {ω^2}sin(ω t- θ)dt$$

Substituting these values in equation given at the beginning-

$$-A {ω^2}sin (ω t- θ) + 2bAw cos (ω t- θ) + {n^2}A sin (ω t- θ) = f sin (ω t- θ + θ)$$

rearranging $$A({n^2-w^2}) sin (ω t- θ) + 2bAw cos (ω t- θ) =sin (ω t- θ) cos θ + cos (ω t- θ) sin θ$$

Since this equation is valid for all value of t, therefore by equating the coefficients of sin(ω t- θ) and cos(ω t- θ) separately, we get

$$A ({n^2-w^2}) = f cos θ $$ $$ 2bAw = f sin θ$$

Squaring and adding the above equations-

$$A^2[({n^2-w^2})^2 + 4b^2 ω^2 ] = f^2[cos^2 θ+ sin^2 θ]$$

Putting these values of A and θ in $$x_2 =A$$ and sin (ω t- θ), the solution of equation-

When the damping is zero (i.e., when b = 0), in steady state$$ x = f(n^2-w^2) sin wt$$

So θ = 0° (i.e., the driving force and displacement will be in same phase). It is concluded that the phase difference between the displacement and driving force of a forced oscillator is due to damping.