---
title: "Stiffness, flexibility and reciprocity theorem"
subject: "Theory of Mechanism and Machine II"
chapter: 8
key: "semester_7/theory_of_mechanism_and_machine_ii:matrix-equations-flexibility-reciprocity"
source: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/matrix-equations-flexibility-reciprocity
site: BCT Engineering
---

# Stiffness, flexibility and reciprocity theorem

Theory of Mechanism and Machine II · Chapter 8: Vibration of Multi Degree of Freedom Systems

- Asked in: year not given, 6 marks
- Video: [Influence Coefficients & Maxwell's Reciprocal Theorem | Mechanical Vibration By Deepak sharma](https://www.youtube.com/watch?v=Ihn-ng_qPCY) — Arya College of Engineering and IT
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/theory_of_mechanism_and_machine_ii/matrix-equations-flexibility-reciprocity
- Whole chapter: https://www.bctengineer.com/md/semester_7/theory_of_mechanism_and_machine_ii/chapter-8

## Question

For a three-degree-of-freedom spring-mass system, write the equations of motion in matrix form. Define the stiffness and flexibility influence coefficients, show that $[a] = [k]^{-1}$, and state and prove Maxwell's reciprocity theorem.

## Short answer

Equations: [M]{ẍ}+[K]{x}={F}. Stiffness coefficient kᵢⱼ is force at i for unit displacement at j (others zero); flexibility aᵢⱼ is displacement at i for unit force at j. They satisfy [a]=[k]⁻¹. Maxwell’s reciprocity: aᵢⱼ = aⱼᵢ, proved by equality of strain energy for reversed loading.

## Answer

#### Equations of motion
A system of three masses $m_1, m_2, m_3$ in a line, connected by springs $k_1$ (wall to $m_1$), $k_2$ and $k_3$ in series, has

$$
\begin{bmatrix} m_1 & 0 & 0 \\ 0 & m_2 & 0 \\ 0 & 0 & m_3 \end{bmatrix}\begin{Bmatrix} \ddot{x}_1 \\ \ddot{x}_2 \\ \ddot{x}_3 \end{Bmatrix} + \begin{bmatrix} k_1+k_2 & -k_2 & 0 \\ -k_2 & k_2+k_3 & -k_3 \\ 0 & -k_3 & k_3 \end{bmatrix}\begin{Bmatrix} x_1 \\ x_2 \\ x_3 \end{Bmatrix} = \begin{Bmatrix} F_1 \\ F_2 \\ F_3 \end{Bmatrix}
$$

i.e. $[M]\{\ddot{x}\} + [K]\{x\} = \{F\}$.

#### Influence coefficients
- **Stiffness influence coefficient $k_{ij}$:** the force at coordinate $i$ required to produce a unit displacement at coordinate $j$, with all other displacements held at zero.
- **Flexibility influence coefficient $a_{ij}$:** the displacement at coordinate $i$ due to a unit force at coordinate $j$ (other forces zero).

Therefore $\{F\} = [k]\{x\}$ and $\{x\} = [a]\{F\}$.

#### Relation between them
Substituting $\{x\} = [a]\{F\}$ in $\{F\} = [k]\{x\}$ gives $\{F\} = [k][a]\{F\}$ for every $\{F\}$, so

$$
[k][a] = [I] \;\Rightarrow\; [a] = [k]^{-1}
$$

The flexibility matrix exists only if the system is restrained (no rigid-body motion). For an unrestrained system $[k]$ is singular.

#### Maxwell's reciprocity theorem
**Statement:** The displacement at point $i$ due to a unit load at point $j$ equals the displacement at $j$ due to a unit load at $i$: $a_{ij} = a_{ji}$ (so $[a]$ and $[k]$ are symmetric).

**Proof:** Apply load $P_i$ first and then $P_j$. The work done (strain energy stored) is
$$
U_1 = \tfrac12P_i(a_{ii}P_i) + \tfrac12P_j(a_{jj}P_j) + P_i(a_{ij}P_j)
$$
(The last term: while $P_j$ is being applied, the full load $P_i$, already in place, moves through the extra deflection $a_{ij}P_j$ at $i$ caused by $P_j$.) Now apply the loads in the reverse order:
$$
U_2 = \tfrac12P_j(a_{jj}P_j) + \tfrac12P_i(a_{ii}P_i) + P_j(a_{ji}P_i)
$$
The final state is the same, and the energy stored in a linear elastic system does not depend on the order of loading, so $U_1 = U_2$, giving $a_{ij} = a_{ji}$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The stiffness influence coefficient k_ij is:
   A. displacement at i due to unit force at j
   B. force at i needed for unit displacement at j with all other displacements zero
   C. force at j when all displacements are one
   D. inverse of the mass at i
   Answer: B. k_ij is the force at coordinate i that produces unit displacement at j while the other displacements are held at zero. (Key point: Stiffness influence coefficient)

2. The flexibility influence coefficient a_ij is:
   A. force at i for unit displacement at j
   B. work done by force at i
   C. displacement at i due to unit force at j (others zero)
   D. stiffness divided by mass
   Answer: C. a_ij is the displacement at i per unit force at j, so {x} = [a]{F}. (Key point: Flexibility influence coefficient)

3. How is [a] = [k]^-1 shown?
   A. Substitute {x} = [a]{F} into {F} = [k]{x}, so [k][a] = [I]
   B. Take the transpose of [k]
   C. Set [a] equal to [M]^-1
   D. Differentiate {F} = [k]{x}
   Answer: A. {F} = [k][a]{F} for every {F} means [k][a] = [I]. It requires a restrained system, otherwise [k] is singular. (Key point: Relation [a] = [k]^-1)

4. Maxwell's reciprocity theorem states that:
   A. a_ij = 1/a_ji
   B. a_ii = a_jj
   C. a_ij = a_ji, i.e. the matrices are symmetric
   D. a_ij = -a_ji
   Answer: C. The deflection at i due to a unit load at j equals the deflection at j due to a unit load at i. (Key point: Maxwell's reciprocity)

5. The proof of reciprocity rests on the fact that:
   A. loads are applied simultaneously
   B. strain energy stored in a linear elastic system does not depend on the order of loading
   C. the springs are massless
   D. the damping is zero
   Answer: B. Loading P_i then P_j and the reverse give the same final energy, so U1 = U2 and a_ji = a_ij. (Key point: Proof of reciprocity)

## Related in Theory of Mechanism and Machine II

- [Forced damped two-degree-of-freedom system](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Adamped-forced-two-dof-equations) — Theory of Mechanism and Machine II, chapter 7
- [Three-mass chain frequencies and modes](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Athree-dof-eigenvalues-mode-shapes) — Theory of Mechanism and Machine II, chapter 8
- [Hartnell governor spring force and stiffness](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Ahartnell-governor-spring-stiffness) — Theory of Mechanism and Machine II, chapter 4
- [Matrix iteration for the first mode](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Amatrix-iteration-fundamental-frequency) — Theory of Mechanism and Machine II, chapter 9

## Related in other subjects

- [Maxwell's equations in phasor form](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Amaxwell-equations-phasor-free-space) — Electromagnetism, chapter 3, asked 2×
- [State and prove Maxwell's reciprocal theorem](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Amaxwell-reciprocal-theorem) — Theory of Structures II, chapter 2
- [Flexibility, stiffness; properties of flexibility matrix](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aflexibility-stiffness-properties-flexibility-matrix) — Theory of Structures II, chapter 2
- [Two-mass system equations, X2/F, F-I and F-V analogies](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ade-tf-fi-fv-m1-m2) — Control Systems, chapter 2