Modal analysis studies how mechanical and structural systems vibrate at their natural frequencies and in their characteristic mode shapes. College engineering students use it to predict resonance, reduce vibration, design safer structures, and interpret test data. This cheat sheet summarizes the core equations and modeling ideas needed for single-degree and multi-degree systems.
It also connects mathematical eigenvalue solutions with physical vibration behavior.
Key Facts
- For an undamped single-degree-of-freedom system, the natural circular frequency is omega_n = sqrt(k/m), where k is stiffness and m is mass.
- The natural frequency in hertz is f_n = omega_n/(2 pi), so a system with omega_n in rad/s must be divided by 2 pi to get cycles per second.
- The damping ratio is zeta = c/c_c = c/(2 sqrt(km)), where c is damping and c_c is the critical damping coefficient.
- For a free undamped multi-degree system, the equation of motion is M x_ddot + K x = 0.
- The modal eigenvalue problem is (K - omega^2 M) phi = 0, where omega is a natural circular frequency and phi is the corresponding mode shape.
- Nontrivial mode shapes exist only when det(K - omega^2 M) = 0.
- With proportional damping, modal superposition can decouple the system response as x(t) = Phi q(t), where Phi is the mode shape matrix and q(t) contains modal coordinates.
- Resonance occurs when an excitation frequency is close to a natural frequency, causing large response amplitudes unless damping or design changes limit the motion.
Vocabulary
- Natural Frequency
- A frequency at which a system tends to vibrate freely after being disturbed.
- Mode Shape
- The relative deformation pattern associated with a specific natural frequency.
- Damping Ratio
- A dimensionless measure of damping compared with the amount needed for critical damping.
- Eigenvalue
- In modal analysis, the value omega^2 that satisfies the vibration eigenvalue equation for a system.
- Modal Superposition
- A method that represents total vibration response as a sum of individual modal responses.
- Resonance
- A condition in which forcing near a natural frequency produces unusually large vibration amplitudes.
Common Mistakes to Avoid
- Using f_n = sqrt(k/m) instead of omega_n = sqrt(k/m) is wrong because sqrt(k/m) gives rad/s, not hertz. Convert with f_n = omega_n/(2 pi).
- Ignoring units in mass and stiffness is wrong because kg, N/m, rad/s, and Hz must be consistent for frequency calculations to be meaningful.
- Treating mode shape magnitudes as absolute displacements is wrong because mode shapes usually show relative motion and can be scaled without changing the mode.
- Assuming every high vibration response is resonance is wrong because large motion can also come from transient loading, low damping, base motion, or poor boundary conditions.
- Using undamped formulas when damping is significant is wrong because damping changes response amplitude and slightly changes damped natural frequency.
Practice Questions
- 1 A 12 kg machine mount has stiffness k = 4800 N/m. Find omega_n in rad/s and f_n in Hz for the undamped single-degree system.
- 2 A system has m = 5 kg, k = 2000 N/m, and c = 40 N s/m. Find the damping ratio zeta.
- 3 For a two-degree system, the determinant equation gives omega^2 = 100 and omega^2 = 400 in rad^2/s^2. Find the two natural circular frequencies and their frequencies in Hz.
- 4 Explain why changing a structure's stiffness, mass distribution, or damping can reduce resonance risk without necessarily eliminating vibration completely.
Understanding Modal Analysis & Natural Frequencies
A vibrating structure stores energy in two forms. When it bends, stretches, or twists, elastic parts store strain energy. When it moves, its mass stores kinetic energy.
Motion continually trades energy between these forms. A stiff, light part changes direction quickly, so its vibration rate is higher. A flexible, heavy part moves more slowly.
This is why a thin ruler flicked from a desk vibrates differently when more of it hangs over the edge. Changing the support condition can matter just as much as changing the material.
Mode shapes show relative motion, not usually actual motion size. In one mode, some locations move strongly while other locations remain nearly still. These nearly still locations are called nodes.
A bridge deck, machine frame, guitar string, or aircraft panel can have many modes, with increasingly complex bending patterns at higher frequencies. The first mode is often the easiest to excite, but it is not always the most dangerous. A higher mode can create large local stress near a joint, hole, weld, or attachment even when the overall movement looks small.
The matrix model depends on choices made before any calculation begins. Engineers divide a real structure into masses, springs, beams, or finite elements. The mass matrix represents where inertia is placed.
The stiffness matrix represents resistance to deformation. Supports must be modeled carefully because a loose bolt, rubber mount, or flexible foundation changes the effective boundary condition.
A model with perfectly fixed supports may predict frequencies that are too high for the real object. Adding mass, such as wiring, fluid, passengers, or attached equipment, commonly lowers frequencies.
Mode shapes have an important mathematical property called orthogonality. Under suitable modeling conditions, one mode can be treated separately from another after the physical coordinates are transformed into modal coordinates. Each modal coordinate behaves like a simpler oscillator with its own frequency and damping.
This makes a large structural problem manageable. In practice, engineers often retain only modes within the frequency range of interest.
That shortcut works only when omitted higher modes have little effect on the measured location or applied load. A force near a node of one mode may excite that mode weakly, while the same force at another location may excite it strongly.
Damping removes vibration energy through mechanisms such as material friction, air resistance, fluid motion, rubbing surfaces, and viscoelastic mounts. Real damping is rarely perfectly proportional, so modes may not separate as neatly as textbook models suggest. Test data can reveal this through frequency response measurements.
During a hammer test or shaker test, sensors record motion while the input force is known. Peaks indicate resonant behavior, but nearby modes, measurement noise, sensor mass, and poor support conditions can distort the result.
Students should track units carefully, distinguish cycles per second from circular frequency, and check whether a result is physically reasonable. A calculated mode with rigid body motion may indicate an unconstrained direction rather than a true flexible vibration.