The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all parts of the system move sinusoidally with that same frequency).
If the oscillating system is driven by an external force at the frequency at which the amplitude of its motion is greatest (close to a natural frequency of the system), this frequency is called resonant frequency.
Free vibrations of an elastic body are called natural vibrations and occur at a frequency called the natural frequency. Natural vibrations are different from forced vibrations which happen at the frequency of an applied force (forced frequency). If the forced frequency is equal to the natural frequency, the vibrations' amplitude increases manyfold. This phenomenon is known as resonance.
In a mass-spring system, with mass m and spring stiffness k, the natural frequency can be calculated as:
In electrical circuits, s1 is a natural frequency of variable x if the zero-input response of x includes the term , where is a constant dependent on initial state of the circuit, network topology, and element values. In a network, sk is a natural frequency of the network if it is a natural frequency of some voltage or current in the network. Natural frequencies depend only on network topology and element values but not the input. It can be shown that the set of natural frequencies in a network can be obtained by calculating the poles of all impedance and admittance functions of the network. All poles of the network transfer function are also natural frequencies of the corresponding response variable; however there may exist some natural frequencies that are not a pole of the network function. These frequencies happen at some special initial states.