Dynamics: Undamped Free Vibration

Understanding undamped free vibration is fundamental to predicting how structures and mechanical components behave when disturbed and left to oscillate on their own. From the suspension of a car to the sway of a skyscraper, the principles governing this simple harmonic motion form the bedrock of engineering dynamics and vibration analysis. Mastering this concept enables you to calculate critical parameters like natural frequency, which dictates a system's inherent speed of oscillation and its susceptibility to resonant failure.

Derivation of the Equation of Motion

The analysis begins with a simple model: a mass $m$ attached to a linear spring with stiffness $k$, resting on a frictionless surface. This is the classic single-degree-of-freedom system. When the mass is displaced from its equilibrium position by a distance $x$, the spring exerts a restoring force proportional to the displacement, given by Hooke's Law: $F_s=-kx$. The negative sign indicates the force always acts opposite to the displacement, pulling the mass back toward equilibrium.

Applying Newton's second law, $F=ma$, and recognizing that acceleration is the second derivative of displacement ($a=\ddot{x}$), we set the net force equal to mass times acceleration. For the free vibration case (no external force) and assuming no damping, the net force is just the spring force: $$m\ddot{x}=-kx$$ Rearranging terms gives the standard form of the equation of motion for undamped free vibration: $$m\ddot{x}+kx=0$$ Dividing the entire equation by the mass $m$ leads to a cleaner form: $$\ddot{x}+\frac{k}{m}x=0$$ This is a homogeneous, second-order linear ordinary differential equation with constant coefficients. Its solution describes the oscillatory motion of the system.

Natural Frequency, Period, and the General Solution

The coefficient of the $x$ term in the simplified equation of motion is a crucial property. We define the square of the natural frequency, denoted ${\omega }_n$, as: $${\omega }_n^2=\frac{k}{m}$$ Therefore, the natural frequency is ${\omega }_n=\sqrt{k/m}$, with units of radians per second (rad/s). It represents the constant angular frequency at which the system will naturally oscillate. A stiffer spring (larger $k$) or a lighter mass (smaller $m$) results in a higher natural frequency—a faster vibration.

The period $T$, which is the time for one complete oscillation cycle, is inversely related to the natural frequency: $$T=\frac{2\pi }{{\omega }_n}=2\pi \sqrt{\frac{m}{k}}$$ The general solution to the differential equation $\ddot{x}+{\omega }_n^{2}x=0$ can be expressed as a combination of sine and cosine functions: $$x(t)=A\cos ({\omega }_{n}t)+B\sin ({\omega }_{n}t)$$ Here, $A$ and $B$ are constants of integration whose values are determined by the initial conditions of the system. This equation describes simple harmonic motion: a sinusoidal oscillation that continues indefinitely in the absence of damping.

Amplitude, Phase Angle, and Applying Initial Conditions

The general solution can be rewritten using a single trigonometric function by introducing an amplitude $X$ and a phase angle $\varphi$. Using a trigonometric identity, we get: $$x(t)=X\cos ({\omega }_{n}t-\varphi )$$ In this form, $X$ represents the maximum displacement of the mass—the peak amplitude of the vibration. The phase angle $\varphi$ determines where in the cycle the oscillation starts at time $t=0$.

To find $X$ and $\varphi$ (or equivalently, $A$ and $B$), you must apply the initial conditions, typically the initial displacement $x(0)=x_0$ and the initial velocity $\dot{x}(0)=v_0$. Substituting $t=0$ into the general solution and its derivative allows you to solve for the constants:

• From $x(0)=A\cos (0)+B\sin (0)=A$, we find $A=x_0$.
• From $\dot{x}(0)=-{\omega }_{n}A\sin (0)+{\omega }_{n}B\cos (0)={\omega }_{n}B$, we find $B=v_0/{\omega }_n$.

The amplitude and phase angle are then found from these constants: $$X=\sqrt{A^2+B^2}=\sqrt{x_0^2+{\left(\frac{v_0}{{\omega }_n}\right)}^2}$$ $$\varphi ={\tan }^{-1}\left(\frac{B}{A}\right)={\tan }^{-1}\left(\frac{v_0}{{\omega }_n{x}_0}\right)$$

The Energy Method for Finding Natural Frequency

An alternative, often simpler, approach to deriving the natural frequency is the energy method. It is based on the conservation of total mechanical energy in an undamped system. The total energy $E$ is the sum of kinetic energy $T$ and potential energy $V$, and it remains constant: $$E=T+V=\text{constant}$$ For the spring-mass system, $T=\frac{1}{2}m{\dot{x}}^2$ and $V=\frac{1}{2}k{x}^2$. Assuming the motion is harmonic, $x(t)=X\cos ({\omega }_{n}t-\varphi )$, we can substitute into the energy expression. At the point of maximum displacement (amplitude $X$), the velocity is zero, so all energy is potential: $E=\frac{1}{2}k{X}^2$. At the equilibrium position ($x=0$), the velocity is maximum (${\dot{x}}_{max}={\omega }_{n}X$), so all energy is kinetic: $E=\frac{1}{2}m({\omega }_{n}X{)}^2$.

Setting these two expressions for constant $E$ equal yields: $$\frac{1}{2}k{X}^2=\frac{1}{2}m{\omega }_n^2{X}^2$$ Solving for ${\omega }_n$ gives the now-familiar result ${\omega }_n=\sqrt{k/m}$. This method is exceptionally powerful for complex systems where deriving the equation of motion might be cumbersome, as you only need expressions for maximum kinetic and potential energy.

Equivalent Spring Constants and Engineering Applications

Real engineering systems rarely involve a single, simple spring. Springs may be arranged in combinations that must be reduced to a single equivalent spring constant $k_{eq}$ for analysis. Two primary configurations exist:

• Springs in Parallel: The equivalent stiffness is the sum of individual stiffnesses: $k_{eq}=k_1+k_2+...$. Parallel springs share the same displacement.
• Springs in Series: The reciprocal of the equivalent stiffness is the sum of reciprocals: $\frac{1}{k_{eq}}=\frac{1}{k_1}+\frac{1}{k_2}+...$. Series springs share the same force.

These concepts are directly applied to engineering vibration problems. For instance, the natural frequency of a vehicle's suspension system can be modeled using equivalent springs for the wheel assemblies. In structural engineering, the lateral stiffness of a simplified building frame can be calculated, and its natural frequency found using ${\omega }_n=\sqrt{k_{eq}/m_{total}}$, providing a first estimate of how the building will sway under wind or seismic loads. The primary goal is often to ensure the operating frequency of any imposed forces is far removed from the structure's natural frequency to avoid destructive resonance.

Common Pitfalls

1. Misidentifying Parallel vs. Series Combinations: A common error is to confuse how springs are connected mechanically with their mathematical combination. Remember the key: if the deflection (displacement) across each spring is necessarily the same, they are in parallel. If the force transmitted through each spring is necessarily the same, they are in series. Draw a free-body diagram to check.
2. Incorrectly Applying Initial Conditions: Students often mismatch the initial velocity $v_0$ with the correct sign when solving for constants. Velocity is the derivative of displacement: $\dot{x}(0)=v_0$. If the mass is given an initial push in the positive x-direction, then $v_0$ is positive. Ensure you substitute into the derivative of your chosen general solution form correctly.
3. Forgetting the Units of Natural Frequency: The formula ${\omega }_n=\sqrt{k/m}$ yields natural frequency in radians per second (rad/s). It is not in Hertz (Hz), which is cycles per second. The relationship is $f_n={\omega }_n/(2\pi )$. Using ${\omega }_n$ where $f_n$ is required, or vice versa, will lead to incorrect calculations for period or response.
4. Sign Error in the Restoring Force: When deriving the equation of motion from Newton's second law, the spring force must be written as $-kx$. Omitting the negative sign invalidates the model, as it no longer represents a restoring force that drives simple harmonic motion.

Summary

• The equation of motion for an undamped, single-degree-of-freedom spring-mass system is $m\ddot{x}+kx=0$, leading to simple harmonic motion described by $x(t)=X\cos ({\omega }_{n}t-\varphi )$.
• The system's intrinsic natural frequency is ${\omega }_n=\sqrt{k/m}$ (rad/s), and its period of oscillation is $T=2\pi /{\omega }_n$. These depend only on the system's physical properties (stiffness and mass), not on the initial disturbance.
• The amplitude $X$ and phase angle $\varphi$ of the oscillation are determined entirely by the initial conditions (initial displacement $x_0$ and initial velocity $v_0$).
• The energy method, based on conservation of total mechanical energy, provides an efficient alternative for finding ${\omega }_n$, especially for complex systems, by equating maximum kinetic and potential energy.
• For practical engineering vibration problems, complex spring arrangements are analyzed by calculating an equivalent spring constant ($k_{eq}$) for parallel or series combinations, which is then used in the natural frequency formula.