Forced Vibration

Forced vibration is the response of a system to a continuing external excitation, usually harmonic in nature. Refer to free vibration for the fundamentals of vibration analysis.

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Harmonic Force on a Mass

Derived from principles of kinetics of rigid bodies, given a harmonic (sinusoidal) external force acting upon a mass-spring-damper system, we can equate the internal and external forces:
$m\ddot{x}+c\dot{x}+kx=F_0\sin \omega t$

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The full solution is a sum of two distinct parts; the complementary solution and the particular solution.

The complementary solution represents the free vibration of the system. It decays exponentially if damping is present ($c>0$). It is known as the transient response as its motion becomes negligible outside the transient phase (beginning phase).

The particular solution represents the long-term, sustained vibration forced onto the system by the external forces ($F_0\sin \omega t$). It is also known as the steady-state response, as it matters long term.
$x=X\sin (\omega t-\varphi )$

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$X$ and $\varphi$ are amplitude and phase lag respectively.

Driving Frequency

Unlike free vibration where an object will vibrate at its natural frequency (${\omega }_n$), it is forced to vibrate at the driving frequency ($\omega$) of the external force.

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Undamped Systems

An undamped system has no energy loss ($c=0$). The steady-state amplitude $X$ is given by:
$X=\frac{F_0}{k-m{\omega }^2}=\frac{F_0/k}{1-r^2}$

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$F_0/k$ is the static deflection which is how far the spring would stretch if $F_0$ was applied statically.

The frequency ratio ($r$) is defined by the following equation:
$r=\frac{\omega }{{\omega }_n}$

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Damped Systems

A damped system needs to account for energy dissipation and phase shift.

Amplitude ($X$)
$X=\frac{F_0}{\sqrt{(k-m{\omega }^2{)}^2+(c\omega {)}^2}}$
Phase Angle ($\varphi$)
$\varphi ={\tan }^{-1}\left(\frac{c\omega }{k-m{\omega }^2}\right)$

Quadrant Selection when $k<m{\omega }^2$

When $k<m{\omega }^2$ (i.e. frequency ratio $r>1$), the denominator of the equation above becomes negative. Thus, the equation will return the wrong quadrant.

To fix this, we must add $180°$ (or $\pi$ radians) to get the correct phase angle between $90°$ and $180°$.

Alternatively, the $\text{atan}2$ function can be used, which accounts for the sign of both $y$ and $x$ to return a value in the correct quadrant.

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Magnification Factor

The magnification factor ($M$) is a ratio of the size of the vibration peak ($X$) relative to the static deflection ($F_0/k$).
$M=\frac{X}{F_0/k}=\frac{1}{\sqrt{(1-r^2{)}^2+(2\zeta r{)}^2}}$

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$\zeta$ is the damping ratio.

Resonance

When the driving frequency is close to the natural frequency ($\omega \approx {\omega }_n$), where the frequency ratio $r\approx 1$, the magnification factor $M$ becomes very large. This is known as resonance.

If there is no damping, then $M\to \infty$ at resonance.

The frequency causing resonance is the natural frequency, but it is also called in this context the critical driving frequency (${\omega }_c={\omega }_n$).

The frequency ratio ($r$) also describes the behaviour of the system:

Frequency Ratio	Phase Relationship	Amplitude Behaviour
$r<1$	In phase with the force.	Follows the forcing function.
$r>1$	$\approx 180°$ out of phase.	Decreases as $r$ increases.
$r\gg 1$	$\approx 180°$ out of phase.	Falls toward zero (mass cannot follow force).

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Support Excitation

Base/support excitation refers to when the support structure an object is resting on or attached to is shaking/vibrating causing the object to experience a forced vibration. Harmonic base motion ($x_B$) is given by:
$x_B=b\sin \omega t$

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$b$ is the base amplitude, which is how far the support moves.

As the base is moving, the spring and damper are no longer attached to a stationary wall. Therefore, the forces they exert depend on relative motion. The spring force and damper force respectively are given by:
$k(x-x_B)c(\dot{x}-{\dot{x}}_B)$

This gives us the following equation of motion when expanded:
$m\ddot{x}+c\dot{x}+kx=kb\sin (\omega t)+c\omega b\cos (\omega t)$

Undamped Support Excitation

For a negligible damper, the system acts as if a force $F_0=kb$ was acting upon it. This leads to a simplified steady-state amplitude formula:
$X=\frac{b}{|1-r^2|}(c\approx 0)$

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Force Transmitted to Base

A vibrating object will exert a force on the foundation it is standing on. The total force ($F_T$) transmitted is the combined spring and damper force.
$F_T=X\sqrt{k^2+(c\omega {)}^2}$

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$X$ is the displacement amplitude of the vibrating mass.

Spring and damper force are out of phase (as one depends on position and the other on velocity). Thus we combine them using a vector sum.

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Unbalanced Rotating Mass

An unbalanced mass ($m_0$) stuck at a distance $e$ (eccentricity) from the centre of a spinning object generates a centrifugal force the acts as a harmonic driving force. It has the following magnitude:
$F_0=m_{0}e{\omega }^2$

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The faster the object spins, the greater the effect of the centrifugal force exerted by the unbalanced mass.

The equation above can be substituted into the damped amplitude formula to get the vibration amplitude $X$ for an unbalanced rotor:
$X=\frac{m_{0}e{\omega }^2}{\sqrt{(k-m{\omega }^2{)}^2+(c\omega {)}^2}}$

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As $X$ increases, the object will physically ‘shake’ back and forth over a larger distance.

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