Vibration-based fault diagnosis (FD) with a simple spectrum can be complex, especially when considering FD of rotating machinery with multiple bearings like a multi-stage turbine. Various studies have sought to better interpret fault spectra, but the process remains equivocal. Consequently, it has been accepted that the simple spectra requires support from additional techniques, such as orbit analysis. But even orbit analysis can be inconclusive. Though promising, attempts at developing viable methods that rival the failure coverage of spectrum analysis without gaining computational complexity remain protracted. Interestingly, few researchers have developed FD methods for transient machine operation, however, these have proven to be involved. Current practices limit vibration data to a single machine, which usually requires a large unique data history. However, if sharing of data between similar machines with different foundations was possible, the need for unique histories would be mitigated. From readily available works, this has not been encountered. Therefore, a simple but robust vibration-based approach is warranted. In light of this, a novel on-bearing vibration response integration approach for condition monitoring of shaft-related faults irrespective of speed and foundation type is proposed in the present study. Vibration data are acquired at different speeds for: a baseline, unbalance, bow, crack, looseness, misalignment, and rub conditions on three laboratory rigs with dynamically different foundations, namely: rigid, flexible support 1 (FS1) and flexible support 2 (FS2). Testing is done on the rigid rig set up first, then FS1, and afterwards FS2. Common vibration features are computed from the measured data to be input to the proposed approach for further processing. First, the proposed approach is developed through its application to a machine at a steady speed in a novel Single-speed FD technique which exploits a single vibration sensor per bearing and fusion of features from different bearings for FD. Initially, vibration features are supplemented with bearing temperature readings with improved classification compared to vibration features alone. However, it is observed that temperature readings are insensitive to faults on the FS1 and FS2 rigs, when compared to vibration features, which are standardised for consistent classification on the different rigs tested. Thus, temperature is not included as a final feature. The observed fault classifications on the different rigs at different speeds with the standardised vibration features are encouraging. Thereafter, a novel Unified Multi-speed FD technique that is based on the initial proposed approach and which works by fusion of vibration features from different bearings at different speeds in a single analysis step for FD is proposed. Experiments on the different rigs repeatedly show the novel Multi-speed technique to be suitable for transient machine operation. Then, a novel generic Multi-foundation Technique (also based on the proposed approach) that allows sharing of vibration data of a wide range of fault conditions between two similarly configured machines with similar speed operation but different foundations is implemented to further mitigate data requirements in the FD process. Observations made with the rigs during steady and transient speed tests show this technique is applicable in situations where data history is available on one machine but lacking on the other. Comparison of experimental results with results obtained from theoretical simulations indicates the approach is consistent. Thus, the proposed approach has the potential for practical considerations.